Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.46.0-wmf.26 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 100 6595 2807093 2802302 2026-04-30T09:15:47Z ~2026-26323-77 3069436 2807093 wikitext text/x-wiki Bună ziua, Am identificat în zona dumneavoastră mai multe firme care ar putea fi interesate de serviciile/produsele dumneavoastră. Practic, clienți potențiali pe care poate nu i-ați luat în calcul până acum. Ce vă pun la dispoziție, pentru doar cateva sute de lei: O extragere din Google Maps a tuturor firmelor din aria dumneavoastră (cu telefon, email, site, adresă), sortate pe categorii de activitate. Datele sunt livrate într-un fișier pe care îl păstrați și îl puteți folosi oricând. Ce puteți face cu aceste date: Contactați direct firmele (prin email, telefon, SMS, WhatsApp), sau prin formularul de contact, așa cum v-am contactat eu. Sau le pot contacta eu pentru dumneavoastră. Ca să vă conving de valoarea informațiilor, vă ofer gratuit câteva unelte pentru a începe: scripturi de contact prin formular soft de trimitere e-mailuri soft de contactare prin WhatsApp opțional: soft de trimis SMS-uri sau de apeluri telefonice cu AI Dacă vă place ideea, îmi puteți scrie pe WhatsApp la 0766-465-311 și putem continua discuția. Mulțumesc pentru timpul acordat. nepi3t6guww29l2x2oy8f5jazq4ba1h 2807094 2807093 2026-04-30T09:16:48Z NDG 3006541 Reverted edits by [[Special:Contributions/~2026-26323-77|~2026-26323-77]] ([[User_talk:~2026-26323-77|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2795770 wikitext text/x-wiki {{center top}} <div style="font-size:133%;border:none;margin:0;padding:.1em;color:#000"> <big>Welcome to the School of Information Technology!</big> <br> <small>''Part of the [[Portal:Engineering and Technology|Engineering and Technology]] Portal.'' <br>The School of Information Technology is moderately integrated with the [[School:Computer Science|School of Computer Science]]. </small> </div> {{center bottom}} <div style="float:right; width:100%"> {{Portal:Engineering/box-header|<big></big>|{{FULLPAGENAME}}/Intro}} {{{{FULLPAGENAME}}/Intro}} {{Portal:Engineering/box-footer|}} </div> <div style="float:right; width:100%"> {{Portal:Engineering/box-header|<big>Core Courses</big>|{{FULLPAGENAME}}/Courses}} {{{{FULLPAGENAME}}/Courses}} {{Portal:Engineering/box-footer|}} </div> <!-- <div style="float:right; width:100%"> {{Portal:Engineering/box-header|<big>Advanced Courses</big>|{{FULLPAGENAME}}/RelatedTopics}} {{{{FULLPAGENAME}}/RelatedTopics}} {{Portal:Engineering/box-footer|}} </div> --> <div style="float:right; width:100%"> {{Portal:Engineering/box-header|<big>Talk</big>|{{FULLPAGENAME}}/Talk}} {{{{FULLPAGENAME}}/Talk}} {{Portal:Engineering/box-footer|}} </div> [[Category:Engineering and Technology]] [[Category:Information technology]] [[Category:Wikiversity schools]] b6skdzg4cr7bqk0ffyduara345cmex2 10 8795 2807004 2771048 2026-04-29T15:56:15Z Dick Bos 24466 more effective links / Wikiproject of that name does not exist 2807004 wikitext text/x-wiki <includeonly> :''This template documentation is [[w:Template doc page pattern|transcluded]] from [[{{FULLPAGENAME}}/doc]]'' [<span class="plainlinks">[{{fullurl:{{FULLPAGENAMEE}}/doc|action=edit}} edit]</span>]</includeonly> <!-- EDIT TEMPLATE DOCUMENTATION BELOW THIS LINE --> This template is used to [[w:WP:CITE|cite sources]] in Wikipedia. It is specifically for books. This template replaces the deprecated {{tl|book reference}}. __NOTOC__ == Usage == All fields '''''must''''' be [[lowercase]]. Copy a blank version to use. Remember that the "|" character must be between each field, the fields must be in lowercase, please delete all the fields that are not being used to clear clutter in the edit window, and ISBN should be in the id field and separated from the number with a space. See also the complete [[#Description of fields|description of fields]]. {| class="wikitable" cellpadding="6" ! colspan="3" | Full version (copy and paste text below and delete parameters you don't need) |- width="50%" valign="top" style="vertical-align:top;" | colspan="3" | :<span style="font-family:Courier; "><nowiki>{{cite book |last= |first= |authorlink= |coauthors= |editor= |others= |title= |origdate= |origyear= |origmonth= |url= |format= |accessdate= |accessyear= |accessmonth= |edition= |date= |publisher= |location= |language= |id= |doi = |pages= |chapter= |chapterurl= |quote= }}</nowiki></span> |- width="50%" valign="top" style="vertical-align:top;" ! colspan="3" | Most commonly used fields (or you can use this and not have to delete as much) |- width="50%" valign="top" style="vertical-align:top;" | colspan="3" | :<span style="font-family:Courier; "><nowiki>{{cite book |last= |first= |authorlink= |coauthors= |title= |year= |publisher= |location= |id= }}</nowiki></span> |- width="50%" valign="top" style="vertical-align:top;" ! Example 1 | style="vertical-align: top;" | <span style="font-family:Courier; "><nowiki>{{cite book |last= Cordell |first= Bruce R. |coauthors= Jeff Grubb, David Yu |title= [[Manual of the Planes]] |publisher= [[Wizards of the Coast]] |date=September 2001|isbn= 0-7869-1850-0 }}</nowiki></span> | style="vertical-align: top;" | {{cite book |last= Cordell |first= Bruce R. |coauthors= Jeff Grubb, David Yu |title= [[Manual of the Planes]] |publisher= [[Wizards of the Coast]] |date=September 2001|isbn= 0-7869-1850-0}} |- width="50%" valign="top" style="vertical-align:top;" ! Example 2 | style="vertical-align: top;" |<pre> {{cite book | last = Mumford | first = David | authorlink = David Mumford | title = The Red Book of Varieties and Schemes | publisher = [[Springer-Verlag]] | year = 1999 | doi = 10.1007/b62130 | isbn = 354063293X }}</pre> | style="vertical-align: top;" | {{cite book | last = Mumford | first = David | authorlink = David Mumford | title = The Red Book of Varieties and Schemes | publisher = [[Springer-Verlag]] | year = 1999 | doi = 10.1007/b62130 | isbn = 354063293X }} |- width="50%" valign="top" style="vertical-align:top;" ! Vertical list !! Prerequisites and Brief Instructions | rowspan="2" width="100%" | &nbsp; |- width="50%" valign="top" style="vertical-align:top;" | style="vertical-align:top;padding:1ex;" |<pre> &nbsp; &nbsp; {{cite book | last = | first = | authorlink = | coauthors = | editor = | others = | title = | origdate = | origyear = | origmonth = | url = | format = | accessdate = | accessyear = | accessmonth = | edition = | date = | year = | month = | publisher = | location = | language = | id = | doi = | pages = | chapter = | chapterurl = | quote = }}</pre> | style="vertical-align:top;padding:1ex;" |<pre> Prerequisites* Brief Instructions -------------- ----------------------- &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(no wikilink) last&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (no wikilink) last last &nbsp; &nbsp; REQUIRED** &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;†preferred (no wikilink) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alternative to origdate &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alternative to origdate &nbsp; url url&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;†preferred (no wikilink) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alternative to accessdate &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alternative to accessdate &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;†preferred (no wikilink) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alternative to date &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;alternative to date &nbsp; publisher &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(no wikilink)&Dagger; chapter &nbsp; &nbsp;</pre> |} {| style="background:none; color:inherit;" |- valign="top" | style="text-align:right;" | &lowast; || The field listed below is a prerequisite for the field to the left. For example '''last''' is a prerequisite to '''first''' meaning '''first''' will NOT be displayed even if it has a value unless '''last''' also has a value. |- valign="top" | style="text-align:right;" | &lowast;&lowast; || Wikilinks '''''must''''' link to an existing article, and cannot be used if '''url''' is used below. |- valign="top" | style="text-align:right;" | † || This is the preferred field with its alternates listed below. |- valign="top" | style="text-align:right;" | &Dagger; || If '''chapterurl''' is provided then '''chapter''' can not have wikilinks. |} == Fields == ===Wikilinks=== Most fields can be wikilinked (ie. ''title'' = <nowiki>[[book article|book title]]</nowiki>), but should generally only be linked to an existing Wikipedia article. Any wikilinked field '''must not''' contain any brackets apart from normal round brackets <code>()</code> &mdash; don't use <code><nowiki><>[]{}</nowiki></code>. ===Description of fields=== ====<small>Syntax (for the technical-minded)</small>==== <div style="font-size: 90%;"> Nested fields either rely on their parent fields, or replace them: *''parent'' **''child'' &mdash; may be used '''with''' ''parent'' (and is ignored if ''parent'' is not used) **OR: ''child2'' &mdash; may be used '''instead''' of ''parent'' (and is ignored if ''parent'' is used) </div> ====Description==== * '''last''': Surname of author. Don't wikilink (use ''authorlink'' instead). ** '''first''': First name(s) of author, including title(s) (eg. ''Firstname Middlename'' or ''Firstname M.'' or ''Dr. Firstname M., Snr.''). Don't wikilink (use ''authorlink'' instead). *** The `last' and `first' fields are poorly named for the case of an author whose surname is usually written first (e.g. as in Chinese). They also have the problem of only communicating which is the surname, not communicating where the surname is usually written. Consider deprecating first,last fields, and reinstating author field, using the notation "Smith, John" or "Hu Ke Jie" as appropriate (i.e. always writing surname first, and using comma or not depending on whether the name is usually written surname last or first). ** '''authorlink''': Title of Wikipedia article about author. Article should already exist. Must not be wikilinked itself. Do not use this on its own, but along with "author" or "first" and "last". ** '''coauthors''': Full name of additional author or authors, surname last, separated by ", " (eg. ''Joe Bloggs, John F. Kennedy, H. R. Dent''). *** Someone please confirm that surname is to go last even for people whose name is usually written surname first. Consider rethinking this field, e.g. using ''Bloggs, Joe and Kennedy, John F. and Dent, H. R. and Hu Ke Jie''. ** OR: '''author''': Full name of author, preferably surname first. * '''editor''': Name of editor/editors. No text is added, so labels such as "(ed.)" have to be supplied by the user. * '''others''': For uses such as "illustrated by Smith" or "trans. Smith". * '''title''': Title of book. '''''This is the only required parameter.''''' Can be wikilinked '''only''' to an existing Wikipedia article. Do not use italics. * '''url''': URL of an online book. Cannot be used if you wikilinked ''title''. ** '''format''': Format, e.g. PDF. HTML implied if not specified. ** '''accessdate''': Full date when item was accessed, in [[ISO 8601]] YYYY-MM-DD format, eg. ''2006-02-17''. '''Required when ''url'' field is used.''' Must not be wikilinked. *** OR: '''accessyear''': Year when item was accessed, and '''accessmonth''': Month when item was accessed. If you also have the day, use ''accessdate'' instead. Must not be wikilinked. * '''edition''': When the book has more than one edition. ''eg:'' "2nd edition". * '''origdate''': Full date of publication of original edition, in [[ISO 8601]] YYYY-MM-DD format, eg. ''2004-06-27''. Must not be wikilinked. ** OR: '''origyear''': Year of publication of original edition, and '''origmonth''': Month of publication of original edition. If you also have the day, use ''date'' instead. Must not be wikilinked. * '''date''': Full date of publication, in [[ISO 8601]] YYYY-MM-DD format, eg. ''2006-02-17''. Must not be wikilinked. ** OR: '''year''': Year of publication, and '''month''': Name of the month of publication. If you also have the day, use ''date'' instead. Must not be wikilinked. * '''publisher''': Publisher should not include corporate designation such is "Ltd" or "Inc". ** '''location''': Place of publication. * '''language''': The language the book is written in, if it is not English. * '''id''': Identifier such as ''<nowiki>ISBN 1-111-22222-9</nowiki>'', ''<nowiki>{{</nowiki>[[:Template:LCC|LCC]]|Z253.U69}}'' or ''<nowiki>{{</nowiki>[[:Template:OCLC|OCLC]]|123456789}}''. Remember, you '''must''' specify the kind of identifier, not just give a number. * '''doi''': [[Wikipedia:digital object identifier|digital object identifier]]. See also {{tl|doi}} * '''pages''': ''<nowiki>5&ndash;7</nowiki>'': first page and optional last page. This is for listing the pages relevant to the citation, not the total number of pages in the book. * '''chapter''': The chapter of the book, written in full. Punctuation other than quotes should be included in the value passed to the parameter, eg. ''chapter = Meet Dick and Jane.'' produces "Meet Dick and Jane." ahead of ''title''. ** '''chapterurl''': URL of an individual chapter of online book. Should be at the same site as ''url'', if any. * '''quote''': Relevant quote from the book. ==Examples== ;Just a title: :<code><nowiki>* {{cite book | title=Mysterious book }}</nowiki></code> :* {{cite book | title = Mysterious book }} ;Year and title: :<code><nowiki>* {{cite book | title=Mysterious book | year=1901 }}</nowiki></code> :* {{cite book | title = Mysterious book | year = 1901 }} ;Basic usage: :<code><nowiki>* {{cite book | first=Joe | last=Bloggs | authorlink=Joe Bloggs | year=1974 | title=Book of Bloggs }}</nowiki></code> :* {{cite book | first=Joe | last=Bloggs | authorlink=Joe Bloggs | year=1974 | title=Book of Bloggs }} ;Basic usage with url: :<code><nowiki>* {{cite book | last=Bloggs | first=Joe | authorlink=Joe Bloggs | year=1974 | title=Book of Bloggs | edition=1st | url=http://en.wikipedia.org/ | accessdate=2006-02-17 }}</nowiki></code> :* {{cite book | last=Bloggs | first=Joe | authorlink=Joe Bloggs | year=1974 | title=Book of Bloggs | edition=1st | url=http://en.wikipedia.org/ | accessdate=2006-02-17 }} ;Three authors, title with a piped wikilink, edition :<code><nowiki>* {{cite book | last=Bloggs | first=Joe | authorlink=Joe Bloggs | coauthors=John Smith, Jim Smythe | title=[[A Thousand Acres|1000 Acres]] | edition=2nd }}</nowiki></code> :* {{cite book | last=Bloggs | first=Joe | authorlink=Joe Bloggs | coauthors=John Smith, Jim Smythe | title=[[A Thousand Acres|1000 Acres]] | edition=2nd }} ;Date without day, wikilinked title and publisher, id, pages, location :<code><nowiki>* {{cite book | last=Cordell | first=Bruce R. | coauthors=Jeff Grubb, David Noonan |date=September 2001| title=[[Manual of the Planes]] | publisher=[[Wizards of the Coast]] | location=Timbuktu | isbn=0-7869-1850-0 | pages=134-137 }}</nowiki></code> :* {{cite book | last=Cordell | first=Bruce R. | coauthors=Jeff Grubb, David Noonan |date=September 2001| title=[[Manual of the Planes]] | publisher=[[Wizards of the Coast]] | location=Timbuktu | isbn=0-7869-1850-0 | pages=134-137 }} ;Date of first edition, other language, illustrator :<code><nowiki>* {{cite book | last=Bloggs | first=Joe | origyear=1463 | year=1974 | title=Book of Bloggs | edition=1st | others=illustrated by Smith | language=German | url=http://en.wikipedia.org/ | accessdate=2006-02-17 }}</nowiki></code> :* {{cite book | last=Bloggs | first=Joe | origyear=1463 | year=1974 | title=Book of Bloggs | edition=1st | others=illustrated by Smith | language=German | url=http://en.wikipedia.org/ | accessdate=2006-02-17 }} ;Using a [[Wikipedia:Digital object identifier|DOI]] :<code><nowiki>*{{cite book | last=Mumford | first=David | authorlink=David Mumford | year=1999 | title=The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians | edition=2nd | publisher=[[Springer-Verlag]] | doi=10.1007/b62130 | isbn=354063293X }}</nowiki></code> :*{{cite book | last=Mumford | first=David | authorlink=David Mumford | year=1999 | title=The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians | edition=2nd | publisher=[[Springer-Verlag]] | doi=10.1007/b62130 | isbn=354063293X }} ==Testing== See [[Template:cite book/regression tests]]. ==Note== Note the extra full-stop when the last author ends with an initial, and there is no date: *{{cite book | last = Invisible | first = M. | title = Mysterious book }} We don't know of a practical solution to this &mdash; unless there is a way to test the characters of a field? ==Citation styles== Established citation styles for coauthors: * [http://www.english.uiuc.edu/cws/wworkshop/writer_resources/citation_styles/mla/mla.htm MLA style]: Last, First and First Last. "If there are more than three authors, you may list only the first author followed by the phrase ''et al''." * [http://www.english.uiuc.edu/cws/wworkshop/writer_resources/citation_styles/apa/apa.htm APA style]: Last, F. & Last, F. * [http://www.liunet.edu/cwis/cwp/library/workshop/citchi.htm Chicago Manual of Style]: Last, First, and First Last. * [http://www.liunet.edu/cwis/cwp/library/workshop/cittur.htm Turabian]: same as Chicago Reference List, above. * [http://www.library.uq.edu.au/training/citation/harvard.html Harvard]: Last, F., Last, F. & Last, F. ==TemplateData== {{TemplateDataHeader}} <div style="overflow:auto;"> <templatedata> { "description": "This template formats a citation to a book using the provided bibliographic information (such as author and title) as well as various formatting options.", "params": { "url": { "label": "URL", "description": "The URL of the online location where the text of the publication can be found. Requires schemes of the type \"http://...\" or maybe even the protocol relative scheme \"//...\"", "type": "string", "aliases": [ "URL" ], "example": "https://www.nytimes.com/..." }, "title": { "label": "Title", "description": "The title of the book; displays in italics", "type": "string", "required": true }, "last": { "label": "Last name", "description": "The surname of the author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors", "aliases": [ "last1", "author", "author1", "author1-last", "author-last", "surname1", "author-last1", "subject1", "surname", "author-last", "subject" ], "suggested": true, "type": "string" }, "first": { "label": "First name", "description": "Given or first name, middle names, or initials of the author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors", "aliases": [ "given", "author-first", "first1", "given1", "author-first1", "author1-first" ], "suggested": true, "type": "string" }, "last2": { "label": "Last name 2", "description": "The surname of the second author; don't wikilink, use 'authorlink2'; can suffix with a numeral to add additional authors", "aliases": [ "author2", "surname2", "author-last2", "author2-last", "subject2" ], "type": "string" }, "first2": { "label": "First name 2", "description": "Given or first name, middle names, or initials of the second author; don't wikilink, use 'authorlink'; can suffix with a numeral to add additional authors", "aliases": [ "given2", "author-first2", "author2-first" ], "type": "string" }, "last3": { "label": "Last name 3", "description": "The surname of the third author; don't wikilink, use 'authorlink3'.", "aliases": [ "author3", "surname3", "author-last3", "author3-last", "subject3" ], "type": "string" }, "first3": { "label": "First name 3", "description": "Given or first name, middle names, or initials of the third author; don't wikilink.", "aliases": [ "given3", "author-first3", "author3-first" ], "type": "string" }, "last4": { "label": "Last name 4", "description": "The surname of the fourth author; don't wikilink, use 'authorlink4'.", "aliases": [ "author4", "surname4", "author-last4", "author4-last", "subject4" ], "type": "string" }, "first4": { "label": "First name 4", "description": "Given or first name, middle names, or initials of the fourth author; don't wikilink.", "aliases": [ "given4", "author-first4", "author4-first" ], "type": "string" }, "last5": { "label": "Last name 5", "description": "The surname of the fifth author; don't wikilink, use 'authorlink5'.", "aliases": [ "author5", "surname5", "author-last5", "author5-last", "subject5" ], "type": "string" }, "first5": { "label": "First name 5", "description": "Given or first name, middle names, or initials of the fifth author; don't wikilink.", "aliases": [ "given5", "author-first5", "author5-first" ], "type": "string" }, "last6": { "label": "Last name 6", "description": "The surname of the sixth author; don't wikilink, use 'authorlink6'.", "aliases": [ "author6", "surname6", "author-last6", "author6-last", "subject6" ], "type": "string" }, "first6": { "label": "First name 6", "description": "Given or first name, middle names, or initials of the sixth author; don't wikilink.", "aliases": [ "given6", "author-first6", "author6-first" ], "type": "string" }, "last7": { "label": "Last name 7", "description": "The surname of the seventh author; don't wikilink, use 'authorlink7'.", "aliases": [ "author7", "surname7", "author-last7", "author7-last", "subject7" ], "type": "string" }, "first7": { "label": "First name 7", "description": "Given or first name, middle names, or initials of the seventh author; don't wikilink.", "aliases": [ "given7", "author-first7", "author7-first" ], "type": "string" }, "last8": { "label": "Last name 8", "description": "The surname of the eighth author; don't wikilink, use 'authorlink8'.", "aliases": [ "author8", "surname8", "author-last8", "author8-last", "subject8" ], "type": "string" }, "first8": { "label": "First name 8", "description": "Given or first name, middle names, or initials of the eighth author; don't wikilink.", "aliases": [ "given8", "author-first8", "author8-first" ], "type": "string" }, "last9": { "label": "Last name 9", "description": "The surname of the ninth author; don't wikilink, use 'authorlink9'. If nine authors are defined, then only eight will show and 'et al.' will show in place of the last author.", "aliases": [ "author9", "surname9", "author-last9", "author9-last", "subject9" ], "type": "string" }, "first9": { "label": "First name 9", "description": "Given or first name, middle names, or initials of the ninth author; don't wikilink.", "aliases": [ "given9", "author-first9", "author9-first" ], "type": "string" }, "date": { "label": "Date", "description": "Full date of the source; do not wikilink", "type": "date", "aliases": [ "air-date", "airdate" ] }, "work": { "label": "Work", "description": "Name of the work in which the cited book text is found", "type": "string", "aliases": [ "journal", "website", "newspaper", "magazine", "encyclopedia", "encyclopaedia", "dictionary", "mailinglist" ] }, "publisher": { "label": "Publisher", "description": "Name of the publisher; displays after title", "type": "string", "suggested": true, "aliases": [ "distributor", "institution", "newsgroup" ] }, "others": { "label": "Others", "description": "Used to record other contributions to the work, such as 'Illustrated by John Smith' or 'Translated by John Smith'", "type": "string" }, "year": { "label": "Year of publication", "description": "Year of the source being referenced; use 'date' instead, if month and day are also known", "type": "string", "suggested": true }, "isbn": { "label": "ISBN", "description": "International Standard Book Number; use the 13-digit ISBN where possible", "type": "string", "suggested": true, "aliases": [ "ISBN13", "isbn13", "ISBN" ] }, "editor-last": { "label": "Editor last name", "description": "The surname of the editor; don't wikilink, use 'editor-link'; can suffix with a numeral to add additional editors", "aliases": [ "editor", "editor-surname", "editor-last1", "editor-surname1", "editor1", "editor1-last", "editor1-surname" ] }, "editor-first": { "label": "Editor first name", "description": "Given or first name, middle names, or initials of the editor; don't wikilink, use 'editor-link'; can suffix with a numeral to add additional editors", "aliases": [ "editor-given", "editor-first1", "editor-given1", "editor1-first", "editor1-given" ] }, "editor-link": { "label": "Link for editor", "description": "Title of existing Wikipedia article about the editor", "type": "wiki-page-name", "aliases": [ "editorlink", "editor-link1", "editor1-link", "editorlink1", "editor1link" ] }, "editor-mask": { "label": "Editor mask", "description": "Replaces the name of the first editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask", "editormask1", "editor1-mask", "editor-mask1", "editor1mask" ] }, "edition": { "label": "Edition", "description": "When the publication has more than one edition; for example: '2nd', 'Revised' etc.; suffixed by ' ed.'", "type": "string" }, "series": { "label": "Series identifier", "description": "Series identifier when the source is part of a series, such as a book series or a journal", "aliases": [ "version" ], "type": "string" }, "volume": { "label": "Volume", "description": "For one publication published in several volumes", "type": "string" }, "location": { "label": "Location of publication", "description": "Geographical place of publication; usually not wikilinked; omit when the publication name includes place", "aliases": [ "place" ], "type": "string", "suggested": true }, "publication-place": { "label": "Place of publication", "description": "Publication place shows after title; if 'place' or 'location' are also given, they are displayed before the title prefixed with 'written at'", "aliases": [ "publicationplace" ], "type": "string" }, "publication-date": { "label": "Publication date", "description": "Date of publication when different from the date the work was written; do not wikilink", "type": "string", "aliases": [ "publicationdate" ] }, "page": { "label": "Page", "description": "The number of a single page in the source that supports the content; displays after 'p.'; use either page= or pages=, but not both", "type": "string", "aliases": [ "p" ] }, "pages": { "label": "Page(s) cited", "description": "A range of pages in the source that support the content (not an indication of the number of pages in the source); displays after 'pp.'; use either page= or pages=, but not both", "type": "string", "suggested": true, "aliases": [ "pp" ] }, "nopp": { "label": "No pp", "description": "Set to 'y' to suppress the 'p.' or 'pp.' display with 'page' or 'pages' when inappropriate (such as 'Front cover')", "type": "string" }, "at": { "label": "At", "description": "May be used instead of 'page' or 'pages' where a page number is inappropriate or insufficient", "type": "string" }, "language": { "label": "Language", "description": "The language in which the source is written, if not English; use the full language name; do not use icons or templates", "type": "string", "aliases": [ "in" ] }, "script-title": { "label": "Script title", "description": "For titles in languages that do not use a Latin-based alphabet (Arabic, Chinese, Cyrillic, Greek, Hebrew, Japanese, Korean, Vietnamese, etc). Prefix with two-character ISO639-1 language code followed by a colon. For Japanese use: |script-title=ja:...", "type": "string" }, "trans-title": { "label": "Translated title", "description": "An English language title, if the source cited is in a foreign language; 'language' is recommended", "type": "string", "aliases": [ "trans_title" ] }, "chapter": { "label": "Chapter", "description": "The chapter heading of the source; may be wikilinked or with 'chapterurl' but not both. For the contribution alias, see contributor-last", "type": "string", "aliases": [ "contribution", "entry", "article", "section" ] }, "trans-chapter": { "label": "Translated chapter", "description": "An English language chapter heading, if the source cited is in a foreign language; 'language' is recommended", "type": "string", "aliases": [ "trans_chapter" ] }, "type": { "label": "Type", "description": "Additional information about the media type of the source; format in sentence case", "type": "string", "aliases": [ "medium" ] }, "format": { "label": "Format", "description": "Format of the work referred to by 'url'; examples: PDF, DOC, XLS; do not specify HTML", "type": "string" }, "arxiv": { "label": "arXiv identifier", "description": "An identifier for arXive electronic preprints of scientific papers", "type": "string", "aliases": [ "ARXIV", "eprint" ] }, "asin": { "label": "ASIN", "description": "Amazon Standard Identification Number; 10 characters", "type": "string", "aliases": [ "ASIN" ] }, "asin-tld": { "label": "ASIN TLD", "description": "ASIN top-level domain for Amazon sites other than the US", "type": "string" }, "bibcode": { "label": "Bibcode", "description": "Bibliographic Reference Code (REFCODE); 19 characters", "type": "string" }, "biorxiv": { "label": "biorXiv", "description": "biorXiv identifier; 6 digits", "type": "line" }, "citeseerx": { "label": "CiteSeerX", "description": "CiteSeerX identifier; found after the 'doi=' query parameter", "type": "line" }, "doi": { "label": "DOI", "description": "Digital Object Identifier; begins with '10.'", "type": "string", "aliases": [ "DOI" ] }, "issn": { "label": "ISSN", "description": "International Standard Serial Number; 8 characters; may be split into two groups of four using a hyphen", "type": "string", "aliases": [ "ISSN" ] }, "jfm": { "label": "jfm code", "description": "Jahrbuch über die Fortschritte der Mathematik classification code", "type": "string" }, "jstor": { "label": "JSTOR", "description": "JSTOR identifier", "type": "string", "aliases": [ "JSTOR" ] }, "lccn": { "label": "LCCN", "description": "Library of Congress Control Number", "type": "string", "aliases": [ "LCCN" ] }, "mr": { "label": "MR", "description": "Mathematical Reviews identifier", "type": "string", "aliases": [ "MR" ] }, "oclc": { "label": "OCLC", "description": "Online Computer Library Center number", "type": "string", "aliases": [ "OCLC" ] }, "ol": { "label": "OL", "description": "Open Library identifier; do not include \"OL\" at beginning of identifier", "type": "string", "aliases": [ "OL" ] }, "osti": { "label": "OSTI", "description": "Office of Scientific and Technical Information identifier", "type": "string", "aliases": [ "OSTI" ] }, "pmc": { "label": "PMC", "description": "PubMed Center article number", "type": "string" }, "pmid": { "label": "PMID", "description": "PubMed Unique Identifier", "type": "string", "aliases": [ "PMID" ] }, "rfc": { "label": "RFC", "description": "Request for Comments number", "type": "string" }, "ssrn": { "label": "SSRN", "description": "Social Science Research Network", "type": "string" }, "zbl": { "label": "Zbl", "description": "Zentralblatt MATH journal identifier", "type": "string" }, "id": { "label": "id", "description": "A unique identifier used where none of the specialized ones are applicable", "type": "string", "aliases": [ "ID" ] }, "quote": { "label": "Quote", "description": "Relevant text quoted from the source; displays last, enclosed in quotes; needs to include terminating punctuation", "type": "string", "aliases": [ "quotation" ] }, "ref": { "label": "Ref", "description": "An anchor identifier; can be made the target of wikilinks to full references; special value 'harv' generates an anchor suitable for the harv and sfn templates", "type": "string" }, "name-list-format": { "label": "Name list format", "description": "Accepts the single keyword 'vanc' to emulate Vancouver Style author / editor name-lists.", "type": "string" }, "mode": { "label": "Mode", "description": "Sets separator and terminal punctuation to the style named in the assigned value; allowable values are: 'cs1' or 'cs2'", "type": "string" }, "postscript": { "label": "Postscript", "description": "The closing punctuation for the citation; ignored if 'quote' is defined; to suppress use reserved keyword 'none'", "type": "string", "default": "." }, "registration": { "label": "Registration", "description": "For online sources that require registration, set to 'yes' (or 'y', or 'true'); superseded by subscription if both are set", "type": "string" }, "subscription": { "label": "Subscription", "description": "For online sources that require subscription, set to 'yes' (or 'y', or 'true'); supersedes registration if both are set", "type": "string" }, "author-mask": { "label": "Author mask", "description": "Replaces the name of the first author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask", "authormask1", "author1-mask", "author-mask1", "author1mask" ] }, "author-mask2": { "label": "Author mask 2", "description": "Replaces the name of the second author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask2", "author2-mask", "author2mask" ] }, "author-mask3": { "label": "Author mask 3", "description": "Replaces the name of the third author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask3", "author3-mask", "author3mask" ] }, "author-mask4": { "label": "Author mask 4", "description": "Replaces the name of the fourth author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask4", "author4-mask", "author4mask" ] }, "author-mask5": { "label": "Author mask 5", "description": "Replaces the name of the fifth author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask5", "author5-mask", "author5mask" ] }, "author-mask6": { "label": "Author mask 6", "description": "Replaces the name of the sixth author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask6", "author6-mask", "author6mask" ] }, "author-mask7": { "label": "Author mask 7", "description": "Replaces the name of the seventh author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask7", "author7-mask", "author7mask" ] }, "author-mask8": { "label": "Author mask 8", "description": "Replaces the name of the eighth author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask8", "author8-mask", "author8mask" ] }, "author-mask9": { "label": "Author mask 9", "description": "Replaces the name of the ninth author with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing author separator; for example, 'with' instead", "type": "string", "aliases": [ "authormask9", "author9-mask", "author9mask" ] }, "display-authors": { "label": "Display authors", "description": "number of authors to display before 'et al.' is used", "type": "number", "aliases": [ "displayauthors" ] }, "author-link": { "label": "Author link", "description": "Title of existing Wikipedia article about the author; can suffix with a numeral to add additional authors", "type": "wiki-page-name", "aliases": [ "authorlink", "subjectlink", "subject-link", "authorlink1", "author-link1", "author1-link", "subjectlink1", "author1link", "subject-link1", "subject1-link", "subject1link" ] }, "author-link2": { "label": "Author link 2", "description": "Title of existing Wikipedia article about the second author; can suffix with a numeral to add additional authors", "type": "wiki-page-name", "aliases": [ "authorlink2", "author2-link", "subjectlink2", "author2link", "subject-link2", "subject2-link", "subject2link" ] }, "author-link3": { "label": "Author link 3", "description": "Title of existing Wikipedia article about the third author.", "type": "wiki-page-name", "aliases": [ "authorlink3", "author3-link", "subjectlink3", "author3link", "subject-link3", "subject3-link", "subject3link" ] }, "author-link4": { "label": "Author link 4", "description": "Title of existing Wikipedia article about the fourth author.", "type": "wiki-page-name", "aliases": [ "authorlink4", "author4-link", "subjectlink4", "author4link", "subject-link4", "subject4-link", "subject4link" ] }, "author-link5": { "label": "Author link 5", "description": "Title of existing Wikipedia article about the sixth author.", "type": "wiki-page-name", "aliases": [ "authorlink5", "author5-link", "subjectlink5", "author5link", "subject-link5", "subject5-link", "subject5link" ] }, "author-link6": { "label": "Author link 6", "description": "Title of existing Wikipedia article about the sixth author.", "type": "wiki-page-name", "aliases": [ "authorlink6", "author6-link", "subjectlink6", "author6link", "subject-link6", "subject6-link", "subject6link" ] }, "author-link7": { "label": "Author link 7", "description": "Title of existing Wikipedia article about the seventh author.", "type": "wiki-page-name", "aliases": [ "authorlink7", "author7-link", "subjectlink7", "author7link", "subject-link7", "subject7-link", "subject7link" ] }, "author-link8": { "label": "Author link 8", "description": "Title of existing Wikipedia article about the eighth author.", "type": "wiki-page-name", "aliases": [ "authorlink8", "author8-link", "subjectlink8", "author8link", "subject-link8", "subject8-link", "subject8link" ] }, "author-link9": { "label": "Author link 9", "description": "Title of existing Wikipedia article about the ninth author.", "type": "wiki-page-name", "aliases": [ "authorlink9", "author9-link", "subjectlink9", "author9link", "subject-link9", "subject9-link", "subject9link" ] }, "access-date": { "label": "URL access date", "description": "The full date when the original URL was accessed; do not wikilink", "type": "string", "aliases": [ "accessdate" ] }, "orig-year": { "label": "Original year", "description": "Original year of publication; provide specifics", "type": "string", "aliases": [ "origyear" ] }, "editor-last2": { "label": "Last name of second editor", "description": "The surname of the second editor; don't wikilink, use 'editor2-link'", "type": "string", "aliases": [ "editor-surname2", "editor2", "editor2-last", "editor2-surname" ] }, "editor-first2": { "label": "First name of second editor", "description": "Given or first name, middle names, or initials of the second editor; don't wikilink, use 'editor2-link'", "type": "string", "aliases": [ "editor-given2", "editor2-first", "editor2-given" ] }, "editor-link2": { "label": "Link for second editor", "description": "Title of existing Wikipedia article about the second editor", "type": "wiki-page-name", "aliases": [ "editor2-link", "editorlink2", "editor2link" ] }, "editor-mask2": { "label": "Mask for second editor", "description": "Replaces the name of the second editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask2", "editor2-mask", "editor2mask" ] }, "editor-last3": { "label": "Last name of third editor", "description": "The surname of the third editor; don't wikilink, use 'editor3-link'", "type": "string", "aliases": [ "editor-surname3", "editor3", "editor3-last", "editor3-surname" ] }, "editor-first3": { "label": "First name of third editor", "description": "Given or first name, middle names, or initials of the third editor; don't wikilink, use 'editor3-link'", "type": "string", "aliases": [ "editor-given3", "editor3-first", "editor3-given" ] }, "editor-link3": { "label": "Link for third editor", "description": "Title of existing Wikipedia article about the third editor", "type": "wiki-page-name", "aliases": [ "editor3-link", "editorlink3", "editor3link" ] }, "editor-mask3": { "label": "Mask for third editor", "description": "Replaces the name of the third editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask3", "editor3-mask", "editor3mask" ] }, "editor-last4": { "label": "Last name of fourth editor", "description": "The surname of the fourth editor; don't wikilink, use 'editor4-link'", "type": "string", "aliases": [ "editor-surname4", "editor4", "editor4-last", "editor4-surname" ] }, "editor-first4": { "label": "First name of fourth editor", "description": "Given or first name, middle names, or initials of the fourth editor; don't wikilink, use 'editor4-link'", "type": "string", "aliases": [ "editor-given4", "editor4-first", "editor4-given" ] }, "editor-link4": { "label": "Link for fourth editor", "description": "Title of existing Wikipedia article about the fourth editor", "type": "wiki-page-name", "aliases": [ "editor4-link", "editorlink4", "editor4link" ] }, "editor-mask4": { "label": "Mask for fourth editor", "description": "Replaces the name of the fourth editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask4", "editor4-mask", "editor4mask" ] }, "editor-last5": { "label": "Last name of fifth editor", "description": "The surname of the fifth editor; don't wikilink, use 'editor5-link'", "type": "string", "aliases": [ "editor-surname5", "editor5", "editor5-last", "editor5-surname" ] }, "editor-first5": { "label": "First name of fifth editor", "description": "Given or first name, middle names, or initials of the fifth editor; don't wikilink, use 'editor5-link'", "type": "string", "aliases": [ "editor-given5", "editor5-first", "editor5-given" ] }, "editor-link5": { "label": "Link for fifth editor", "description": "Title of existing Wikipedia article about the fifth editor", "type": "wiki-page-name", "aliases": [ "editor5-link", "editorlink5", "editor5link" ] }, "editor-mask5": { "label": "Mask for fifth editor", "description": "Replaces the name of the fifth editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask5", "editor5-mask", "editor5mask" ] }, "editor-last6": { "label": "Last name of sixth editor", "description": "The surname of the sixth editor; don't wikilink, use 'editor6-link'", "type": "string", "aliases": [ "editor-surname6", "editor6", "editor6-last", "editor6-surname" ] }, "editor-first6": { "label": "First name of sixth editor", "description": "Given or first name, middle names, or initials of the sixth editor; don't wikilink, use 'editor6-link'", "type": "string", "aliases": [ "editor-given6", "editor6-first", "editor6-given" ] }, "editor-link6": { "label": "Link for sixth editor", "description": "Title of existing Wikipedia article about the sixth editor", "type": "wiki-page-name", "aliases": [ "editor6-link", "editorlink6", "editor6link" ] }, "editor-mask6": { "label": "Mask for sixth editor", "description": "Replaces the name of the sixth editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask6", "editor6-mask", "editor6mask" ] }, "editor-last7": { "label": "Last name of seventh editor", "description": "The surname of the seventh editor; don't wikilink, use 'editor7-link'", "type": "string", "aliases": [ "editor-surname7", "editor7", "editor7-last", "editor7-surname" ] }, "editor-first7": { "label": "First name of seventh editor", "description": "Given or first name, middle names, or initials of the seventh editor; don't wikilink, use 'editor7-link'", "type": "string", "aliases": [ "editor-given7", "editor7-first", "editor7-given" ] }, "editor-link7": { "label": "Link for seventh editor", "description": "Title of existing Wikipedia article about the seventh editor", "type": "wiki-page-name", "aliases": [ "editor7-link", "editorlink7", "editor7link" ] }, "editor-mask7": { "label": "Mask for seventh editor", "description": "Replaces the name of the seventh editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask7", "editor7-mask", "editor7mask" ] }, "editor-last8": { "label": "Last name of eighth editor", "description": "The surname of the eighth editor; don't wikilink, use 'editor8-link'", "type": "string", "aliases": [ "editor-surname8", "editor8", "editor8-last", "editor8-surname" ] }, "editor-first8": { "label": "First name of eighth editor", "description": "Given or first name, middle names, or initials of the eighth editor; don't wikilink, use 'editor8-link'", "type": "string", "aliases": [ "editor-given8", "editor8-first", "editor8-given" ] }, "editor-link8": { "label": "Link for eighth editor", "description": "Title of existing Wikipedia article about the eighth editor", "type": "wiki-page-name", "aliases": [ "editor8-link", "editorlink8", "editor8link" ] }, "editor-mask8": { "label": "Mask for eighth editor", "description": "Replaces the name of the eighth editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask8", "editor8-mask", "editor8mask" ] }, "editor-last9": { "label": "Last name of ninth editor", "description": "The surname of the ninth editor; don't wikilink, use 'editor9-link'", "type": "string", "aliases": [ "editor-surname9", "editor9", "editor9-last", "editor9-surname" ] }, "editor-first9": { "label": "First name of ninth editor", "description": "Given or first name, middle names, or initials of the ninth editor; don't wikilink, use 'editor9-link'", "type": "string", "aliases": [ "editor-given9", "editor9-first", "editor9-given" ] }, "editor-link9": { "label": "Link for ninth editor", "description": "Title of existing Wikipedia article about the ninth editor", "type": "wiki-page-name", "aliases": [ "editor9-link", "editorlink9", "editor9link" ] }, "editor-mask9": { "label": "Mask for ninth editor", "description": "Replaces the name of the ninth editor with em dashes or text; set to a numeric value 'n' to set the dash 'n' em spaces wide; set to a text value to display the text without a trailing editor separator; for example, 'with' instead", "type": "string", "aliases": [ "editormask9", "editor9-mask", "editor9mask" ] }, "chapter-url": { "label": "Chapter URL", "description": "The URL of the online location where the text of the chapter can be found", "aliases": [ "chapterurl", "contribution-url", "section-url", "contributionurl", "sectionurl" ], "type": "string" }, "doi-broken-date": { "label": "DOI broken date", "description": "The date that the DOI was determined to be broken", "type": "string", "aliases": [ "doi_brokendate" ] }, "archive-url": { "label": "Archive URL", "description": "The URL of an archived copy of a web page, if or in case the URL becomes unavailable; requires 'archivedate'", "type": "string", "aliases": [ "archiveurl" ] }, "archive-date": { "label": "Archive date", "description": "Date when the original URL was archived; do not wikilink", "type": "string", "aliases": [ "archivedate" ] }, "dead-url": { "label": "Dead URL", "description": "If set to 'no', the title display is adjusted; useful for when the URL is archived preemptively but still live", "type": "string", "aliases": [ "deadurl" ] }, "lay-url": { "label": "Lay URL", "description": "URL link to a non-technical summary or review of the source", "aliases": [ "lay-summary", "laysummary", "layurl" ], "type": "string" }, "lay-source": { "label": "Lay source", "description": "Name of the source of the laysummary; displays in italics, preceded by an en dash", "type": "string", "aliases": [ "laysource" ] }, "lay-date": { "label": "Lay date", "description": "Date of the summary; displays in parentheses", "type": "string", "aliases": [ "laydate" ] }, "last-author-amp": { "label": "Last author ampersand", "description": "When set to any value, changes the separator between the last two names of the author list to 'space ampersand space'", "type": "string", "aliases": [ "lastauthoramp" ] }, "via": { "description": "Aggregate or database provider, when different from the Publisher. Typically used for Ebooks.", "example": "Open Edition, JSTOR ", "type": "string" }, "url-access": { "label": "URL access level", "description": "Classification of the access restrictions on the URL ('registration', 'subscription' or 'limited')", "type": "string" }, "bibcode-access": { "label": "Bibcode access level", "description": "If the full text is available from ADS via this Bibcode, type 'free'.", "type": "string" }, "doi-access": { "label": "DOI access level", "description": "If the full text is free to read via the DOI, type 'free'.", "type": "string" }, "hdl-access": { "label": "HDL access level", "description": "If the full text is free to read via the HDL, type 'free'.", "type": "string" }, "jstor-access": { "label": "Jstor access level", "description": "If the full text is free to read on Jstor, type 'free'.", "type": "string" }, "ol-access": { "label": "OpenLibrary access level", "description": "If the full text is free to read on OpenLibrary, type 'free'.", "type": "string" }, "osti-access": { "label": "OSTI access level", "description": "If the full text is free to read on OSTI, type 'free'.", "type": "string" }, "ismn": { "aliases": [ "ISMN" ], "label": "ISMN", "description": "International Standard Music Number; Use the ISMN actually printed on or in the work. Hyphens or spaces in the ISMN are optional.", "type": "string", "example": "979-0-9016791-7-7" }, "eissn": { "aliases": [ "EISSN" ], "label": "EISSN", "description": "International Standard Serial Number for the electronic media of a serial publication; eight characters may be split into two groups of four using a hyphen, but not an en dash or a space.", "example": "2009-0048", "type": "string" }, "editors": { "label": "Editors list", "description": "Free-form list of editor names; use of this parameter is discouraged", "deprecated": "For multiple editors, use editor-last1, editor-first1 through editor-lastn, editor-firstn" }, "translator-last": { "label": "Translator last name", "description": "The surname of the translator; don't wikilink, use 'translator-link'; can suffix with a numeral to add additional translators.", "aliases": [ "translator", "translator-last1", "translator1", "translator1-last" ], "type": "string" }, "translator-first": { "label": "Translator first name", "description": "Given or first name, middle names, or initials of the translator; don't wikilink, use 'translator-link'; can suffix with a numeral to add additional translators.", "aliases": [ "translator1-first", "translator-first1" ], "type": "string" }, "translator-link": { "label": "Translator link", "description": "Title of existing Wikipedia article about the translator; can suffix with a numeral to add additional translators.", "type": "wiki-page-name", "aliases": [ "translator-link1", "translator1-link" ] }, "translator-last2": { "label": "Translator last name 2", "description": "The surname of the second translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator2", "translator2-last" ], "type": "string" }, "translator-first2": { "label": "Translator first name 2", "description": "Given or first name, middle names, or initials of the second translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator2-first" ], "type": "string" }, "translator-last3": { "label": "Translator last name 3", "description": "The surname of the third translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator3", "translator3-last" ], "type": "string" }, "translator-first3": { "label": "Translator first name 3", "description": "Given or first name, middle names, or initials of the third translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator3-first" ], "type": "string" }, "translator-last4": { "label": "Translator last name 4", "description": "The surname of the fourth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator4", "translator4-last" ], "type": "string" }, "translator-first4": { "label": "Translator first name 4", "description": "Given or first name, middle names, or initials of the fourth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator4-first" ], "type": "string" }, "translator-last5": { "label": "Translator last name 5", "description": "The surname of the fifth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator5", "translator5-last" ], "type": "string" }, "translator-first5": { "label": "Translator first name 5", "description": "Given or first name, middle names, or initials of the fifth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator5-first" ], "type": "string" }, "translator-last6": { "label": "Translator last name 6", "description": "The surname of the sixth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator6", "translator6-last" ], "type": "string" }, "translator-first6": { "label": "Translator first name 6", "description": "Given or first name, middle names, or initials of the sixth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator6-first" ], "type": "string" }, "translator-last7": { "label": "Translator last name 7", "description": "The surname of the seventh translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator7", "translator7-last" ], "type": "string" }, "translator-first7": { "label": "Translator first name 7", "description": "Given or first name, middle names, or initials of the seventh translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator7-first" ], "type": "string" }, "translator-last8": { "label": "Translator last name 8", "description": "The surname of the eighth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator8", "translator8-last" ], "type": "string" }, "translator-first8": { "label": "Translator first name 8", "description": "Given or first name, middle names, or initials of the eighth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator8-first" ], "type": "string" }, "translator-last9": { "label": "Translator last name 9", "description": "The surname of the ninth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator9", "translator9-last" ], "type": "string" }, "translator-first9": { "label": "Translator first name 9", "description": "Given or first name, middle names, or initials of the ninth translator; don't wikilink, use 'translator-link'.", "aliases": [ "translator9-first" ], "type": "string" }, "translator-link2": { "label": "Translator link 2", "description": "Title of existing Wikipedia article about the second translator.", "type": "wiki-page-name", "aliases": [ "translator2-link" ] }, "translator-link3": { "label": "Translator link 3", "description": "Title of existing Wikipedia article about the third translator.", "type": "wiki-page-name", "aliases": [ "translator3-link" ] }, "translator-link4": { "label": "Translator link 4", "description": "Title of existing Wikipedia article about the fourth translator.", "type": "wiki-page-name", "aliases": [ "translator4-link" ] }, "translator-link5": { "label": "Translator link 5", "description": "Title of existing Wikipedia article about the fifth translator.", "type": "wiki-page-name", "aliases": [ "translator5-link" ] }, "translator-link6": { "label": "Translator link 6", "description": "Title of existing Wikipedia article about the sixth translator.", "type": "wiki-page-name", "aliases": [ "translator6-link" ] }, "translator-link7": { "label": "Translator link 7", "description": "Title of existing Wikipedia article about the seventh translator.", "type": "wiki-page-name", "aliases": [ "translator7-link" ] }, "translator-link8": { "label": "Translator link 8", "description": "Title of existing Wikipedia article about the eighth translator.", "type": "wiki-page-name", "aliases": [ "translator8-link" ] }, "translator-link9": { "label": "Translator link 9", "description": "Title of existing Wikipedia article about the ninth translator.", "type": "wiki-page-name", "aliases": [ "translator9-link" ] }, "vauthors": { "label": "Vancouver style author list", "description": "If using Vancouver style, comma separated list of author names; enclose corporate or institutional author names in doubled parentheses", "example": "Smythe JB, ((Megabux Corporation))", "type": "string" }, "issue": { "label": "Issue", "description": "Issue number. This parameter is not supported by and should generally not be used with cite book. Consider that a different cite template may be more appropriate, such as cite magazine or cite journal. See Help:Citation_Style_1#Pages.", "type": "string", "aliases": [ "number" ] }, "coauthor": { "label": "Coauthor", "description": "Coauthor", "type": "string", "deprecated": "Use last# / first# or author or authors. If coauthor · coauthors are used for et al, then use display-authors=etal instead.Include coauthors in author or authors or use separate authorn or lastn/firstn to list coauthors. lastn/firstn is the preferred way of coding this." }, "coauthors": { "label": "Coauthors", "description": "Coauthors", "type": "string", "deprecated": "Use last# / first# or author or authors. If coauthor · coauthors are used for et al, then use display-authors=etal instead.Include coauthors in author or authors or use separate authorn or lastn/firstn to list coauthors. lastn/firstn is the preferred way of coding this." }, "display-editors": { "aliases": [ "displayeditors" ], "label": "Display Editors", "description": "Controls the number of editor names that are displayed when a citation is published. To change the displayed number of editors, set display-editors to the desired number. For example, |display-editors=2 will display only the first two editors in a citation. By default, all editors are displayed. |display-editors=etal displays all editors in the list followed by et al.", "type": "string" }, "authors": { "label": "Authors list", "description": "List of authors as a free form list. Use of this parameter is discouraged, \"lastn\" to \"firstn\" are preferable. Warning: do not use if last or any of its aliases are used.", "type": "string", "aliases": [ "people", "host", "credits" ] }, "veditors": { "label": "Vancouver style editor list", "description": "Comma separated list of editor names in Vancouver style; enclose corporate or institutional names in doubled parentheses", "example": "Smythe JB, ((Megabux Corporation))", "type": "string" }, "city": { "label": "City of Publication", "description": "Where published", "type": "string", "deprecated": "Use place or location instead." }, "chapter-format": { "aliases": [ "contribution-format", "section-format" ], "label": "Format of Chapter URL", "type": "string", "description": "Format of the work referred to by chapter-url; displayed in parentheses after chapter. HTML is implied and should not be specified.", "example": "PDF, DOC, or XLS" }, "agency": { "label": "Agency", "description": "Unusual in cite book. Use if an agency is needed in addition to publisher, etc.", "type": "string" }, "title-link": { "aliases": [ "titlelink", "episode-link", "episodelink" ], "label": "Title link", "description": "Title of existing Wikipedia article about the source named in title – do not use a web address; do not wikilink.", "type": "wiki-page-name" }, "ignore-isbn-error": { "label": "Ignore ISBN Error", "description": "True if ISBN Errors should be ignored.If set, page will be added to a maintenance category \"CS1 maint: Ignored ISBN errors\".", "type": "boolean" }, "collaboration": { "label": "Collaboration", "description": "Name of a group of authors or collaborators; requires author, last, or vauthors which list one or more primary authors; follows author name-list; appends 'et al.' to author name-list.", "type": "string" }, "script-chapter": { "label": "Script Chapter", "description": "Chapter heading for languages that do not use a Latin-based alphabet (Arabic, Chinese, Cyrillic, Greek, Hebrew, Japanese, Korean, Vietnamese, etc); follows transliteration defined in chapter. Should be prefixed with an ISO 639-1 two-character code to help browsers properly display the script", "example": "ja:東京タワー", "type": "string" }, "department": { "label": "Department", "description": "Unusual in cite book." }, "class": { "label": "arXiv Class", "description": "Cite arXiv identifiers", "type": "string" }, "hdl": { "aliases": [ "HDL" ], "label": " Handle System identifier", "description": "Handle System identifier for digital objects and other resources on the Internet", "type": "string" }, "archive-format": { "label": "Archive Format", "description": "Format of the Archive", "type": "string" }, "df": { "label": "Date format", "description": "Sets rendered dates to the specified format", "type": "string" } }, "maps": { "proveit": { "main": "title", "textarea": [ "quote" ] }, "citoid": { "edition": "edition", "title": "title", "bookTitle": "title", "publicationTitle": "title", "url": "url", "publisher": "publisher", "date": "date", "place": "location", "ISSN": [ "issn" ], "ISBN": [ "isbn" ], "oclc": "oclc", "PMCID": "pmc", "PMID": "pmid", "pages": "pages", "volume": "volume", "series": "series", "DOI": "doi", "language": "language", "translator": [ [ "translator-first", "translator-last" ], [ "translator-first2", "translator-last2" ], [ "translator-first3", "translator-last3" ], [ "translator-first3", "translator-last3" ], [ "translator-first4", "translator-last4" ], [ "translator-first5", "translator-last5" ], [ "translator-first6", "translator-last6" ], [ "translator-first7", "translator-last7" ], [ "translator-first8", "translator-last8" ], [ "translator-first9", "translator-last9" ] ], "contributor": "others", "author": [ [ "first", "last" ], [ "first2", "last2" ], [ "first3", "last3" ], [ "first4", "last4" ], [ "first5", "last5" ], [ "first6", "last6" ], [ "first7", "last7" ], [ "first8", "last8" ], [ "first9", "last9" ] ], "editor": [ [ "editor-first", "editor-last" ], [ "editor-first2", "editor-last2" ], [ "editor-first3", "editor-last3" ], [ "editor-first4", "editor-last4" ], [ "editor-first5", "editor-last5" ], [ "editor-first6", "editor-last6" ], [ "editor-first7", "editor-last7" ], [ "editor-first8", "editor-last8" ], [ "editor-first9", "editor-last9" ] ] } }, "paramOrder": [ "url", "title", "title-link", "last", "first", "vauthors", "last2", "first2", "last3", "first3", "last4", "first4", "last5", "first5", "last6", "first6", "last7", "first7", "last8", "first8", "last9", "first9", "collaboration", "date", "work", "publisher", "others", "year", "isbn", "ignore-isbn-error", "editor-last", "editor-first", "editor-link", "editor-mask", "veditors", "editors", "edition", "series", "volume", "location", "publication-place", "publication-date", "page", "pages", "nopp", "at", "language", "translator-last", "translator-first", "script-title", "trans-title", "chapter", "script-chapter", "trans-chapter", "type", "format", "arxiv", "class", "asin", "asin-tld", "bibcode", "biorxiv", "citeseerx", "doi", "eissn", "hdl", "ismn", "issn", "jfm", "jstor", "lccn", "mr", "oclc", "ol", "osti", "pmc", "pmid", "rfc", "ssrn", "zbl", "id", "quote", "ref", "name-list-format", "mode", "postscript", "registration", "subscription", "author-mask", "author-mask2", "author-mask3", "author-mask4", "author-mask5", "author-mask6", "author-mask7", "author-mask8", "author-mask9", "display-authors", "author-link", "author-link2", "author-link3", "author-link4", "author-link5", "author-link6", "author-link7", "author-link8", "author-link9", "access-date", "orig-year", "editor-last2", "editor-first2", "editor-link2", "editor-mask2", "editor-last3", "editor-first3", "editor-link3", "editor-mask3", "editor-last4", "editor-first4", "editor-link4", "editor-mask4", "editor-last5", "editor-first5", "editor-link5", "editor-mask5", "editor-last6", "editor-first6", "editor-link6", "editor-mask6", "editor-last7", "editor-first7", "editor-link7", "editor-mask7", "editor-last8", "editor-first8", "editor-link8", "editor-mask8", "editor-last9", "editor-first9", "editor-link9", "editor-mask9", "display-editors", "translator-last2", "translator-first2", "translator-last3", "translator-first3", "translator-last4", "translator-first4", "translator-last5", "translator-first5", "translator-last6", "translator-first6", "translator-last7", "translator-first7", "translator-last8", "translator-first8", "translator-last9", "translator-first9", "translator-link", "translator-link2", "translator-link3", "translator-link4", "translator-link5", "translator-link6", "translator-link7", "translator-link8", "translator-link9", "chapter-url", "chapter-format", "doi-broken-date", "url-access", "archive-url", "archive-format", "archive-date", "dead-url", "lay-url", "lay-source", "lay-date", "last-author-amp", "via", "bibcode-access", "doi-access", "hdl-access", "jstor-access", "ol-access", "osti-access", "issue", "coauthor", "coauthors", "authors", "city", "agency", "department", "df" ], "format": "inline" } </templatedata> </div> ==See also== * [[w:Wikipedia:Citing sources]]: Guide on Wikipedia * [[w:Wikipedia:Citation templates]]: Citation templates on Wikipedia * [[:Category:Citation templates]] on Wikiversity {{Citation Style 1}} <includeonly> <!-- ADD CATEGORIES BELOW THIS LINE --> [[Category:Citation templates|{{PAGENAME}}]] [[Category:Book templates|{{PAGENAME}}]] [[Category:Templates using ParserFunctions|{{PAGENAME}}]] <!-- ADD INTERWIKIS BELOW THIS LINE --> [[cs:Šablona:Citace knihy]] [[fr:Modèle:Cite book]] [[it:Template:Cita libro]] [[ru:Шаблон:Книга]] [[sv:Mall:Bokref]] </includeonly> 06h42lv2ezeloruexs2kw05st3ig2vy 0 53312 2807083 2806726 2026-04-30T04:17:40Z ~2026-26137-05 3069395 2807083 wikitext text/x-wiki [[File:Alice par John Tenniel 30.png|thumb|right|An illustration from Lewis Carroll's ''Alice's Adventures in Wonderland'', depicting the fictional protagonist, Alice, playing a fantastical game of croquet.]] [[File:AmericasBestComics2901.jpg|thumb]] [[File:Tarzan All Story.jpg|thumb]] [[File:Little Nemo Clowns2.jpg|thumb]] [[File:FairbanksMarkofZorro.jpg|thumb]] Fiction generally is a narrative form, in any medium, consisting of imaginary people, events, or places—in other words, not based strictly on history or fact. It also commonly refers, more narrowly, to written narratives in prose and often specifically novels. In film, it generally corresponds to narrative film in opposition to documentary.<ref>[[Wikipedia: Fiction]]</ref> == Resources == * [[Exploring science through fiction]] * [[Fiction writing support group]] * [[Science Fiction Challenge]] * [[Portal:Literary Studies]] * [[Fiction writing]] == List of fictions == * Cartoons (Mickey Mouse, Winnie the Pooh, Popeye, Betty Boop, Felix the Cat, Looney Tunes and Merrie Melodies, Woody Woodpecker, Rocky and Bullwinkle, The Flintstones, The Jetsons, Scooby-Doo, Mr. Magoo, Mighty Mouse, The Pink Panther, SpongeBob SquarePants, Rugrats, The Loud House, Oggy and the Cockroaches, The Simpsons, Futurama, South Park, Beavis and Butt-Head, King of the Hill, Hazbin Hotel, Johnny Test, Rick and Morty, The Powerpuff Girls, Ben 10, etc.) * Video Games (Mario, Donkey Kong, Sonic the Hedgehog, PaRappa the Rapper, Pac-Man, Mega Man, The Legend of Zelda, Kirby, Crash Bandicoot, Rayman, Banjo Kazooie, Cuphead, Enchanted Portals, Angry Birds, Shantae, etc.) * Comics (Superman, Batman, Wonder Woman, Captain Marvel / Shazam, Spider-Man, The Incredible Hulk, Garfield, The Smurfs, Peanuts, Blondie, Spirou, Marsupilami, The Katzenjammer Kids, Little Nemo in Slumberland, Yakari, Tintin, Heathcliff, Cubitus, Dilbert, Teenage Mutant Ninja Turtles, Rupert Bear, Dennis the Menace (Hank Ketcham), Dennis the Menace and Gnasher, Archie, Baby Blues, Richie Rich, Asterix, Lucky Luke, The Phantom, Buck Rogers, Flash Gordon, etc.) * Anime / Manga (Dragon Ball, Pokémon, Doraemon, Astro Boy, Princess Knight, Maya the Bee, Vicky the Viking, Yu-Gi-Oh!, Sazae-san, Sailor Moon, Beyblade, Robotan, My Neighbor Totoro, FLCL, Tenkai Knights, Attack on Titan, Digimon, Naruto, Bleach, One Piece, Anpanman, etc.) * Films (Star Wars, Indiana Jones, Harry Potter, Underworld, Terminator, Jurassic Park, Kill Bill, The Godfather, Back to the Future, The Chronicles of Narnia, Planet of the Apes, The Lord of the Rings, The Hobbit, Pirates of the Caribbean, Mary Poppins, King Kong, Godzilla, Charlie Chaplin, Laurel and Hardy, The Marx Brothers, Rocky, Mad Max, Fast & Furious, James Bond, Nosferatu, Puppet Master, Mission: Impossible, Gone with the Wind, Police Academy, Jaws, The Exorcist, Saw, A Nightmare on Elm Street, Friday the 13th, Cleopatra, Buster Keaton, Ben-Hur, Child's Play, The Shining, Pulp Fiction, Full Metal Jacket, Men in Black, Don Juan, The Jazz Singer, Lights of New York, The Birth of a Nation, Ted, Titanic, Avatar, Casablanca, Rambo, The Matrix, Halloween, Hellraiser, Home Alone, Marilyn Monroe, Austin Powers, The Wizard of Oz, The Three Stooges, etc.) * Animated Films (Snow White and the Seven Dwarfs, Cinderella, The Little Mermaid, Aladdin, Pinocchio, Hercules, Lilo & Stitch, The Lion King, Gulliver's Travels, Anastasia, etc.) * Computer-Animations (Toy Story, Cars, Luxo Jr., Knick Knack, Tin Toy, Shrek, Ice Age, Frozen, Tangled, Moana, Despicable Me, Miraculous: Tales of Ladybug & Cat Noir, VeggieTales, Antz, A Bug's Life, Wonder Park, Monsters, Inc., Finding Nemo, The Incredibles, etc.) * Stop-Motion Animations (Wallace and Gromit, Shaun the Sheep, Coraline, Pingu, Roary the Racing Car, Bertha, Rudolph the Red-Nosed Reindeer, The Nightmare Before Christmas, Postman Pat, Fireman Sam, Bob the Builder, The Wombles, Gumby, The PJs, Davey and Goliath, etc.) * Literature (Frankenstein, Dracula, The Phantom of the Opera, Sherlock Holmes, Thomas & Friends, Babar the Elephant, Peter Rabbit, Paddington Bear, Where's Wally?, Raggedy Ann and Andy, Dr. Jekyll and Mr. Hyde, Around the World in Eighty Days, The War of the Worlds, Don Quixote, Conan the Barbarian, Nancy Drew, Madeline, The Cat in the Hat, Grinch, Noddy, Tarzan, Mr. Men and Little Miss, Richard Scarry's Busytown, Pride and Prejudice, Alice in Wonderland, Hercule Poirot, Miss Marple, Zorro, 20,000 Leagues Under the Sea, The Three Musketeers, Wuthering Heights, etc.) * Advertising (Ronald McDonald, M&M's, Tony the Tiger, Tetley Tea Folk, Pillsbury Doughboy, Kool-Aid Man, Noid, California Raisins, Jack in the Box, Green Giant, Sonny the Cuckoo Bird, Cap'n Crunch, etc.) * Greeting Cards (Strawberry Shortcake, Care Bears, Rainbow Brite, etc.) * Toys (Transformers, My Little Pony, Barbie, Masters of the Universe, G.I. Joe, etc.) * Radio Series (The Green Hornet, The Lone Ranger, The Archers, etc.) * Television Series (Star Trek, Doctor Who, Mr. Bean, Blackadder, Tugs, Mind Your Language, Monty Python, Between the Lions, The X-Files, Seinfeld, Game of Thrones, Sesame Street, The Muppets, Sam & Cat, The Bill, Benny Hill, Cops, Jackass, Happy Days, The Munsters, The Addams Family, As Time Goes By, Family Ties, Fawlty Towers, ALF, The Dukes of Hazzard, etc.) == See Also == * [[Wikipedia: Fiction]] * [[Wikibooks: Writing Adolescent Fiction]] == References == {{Reflist}} {{subpagesif}} [[Category:Reading]] czto6bhzy0c9wfkmyrjj581ixwrv0at 0 119422 2806985 2806569 2026-04-29T12:06:17Z Young1lim 21186 /* Sequential Circuits */ 2806985 wikitext text/x-wiki == ''' Number Systems '''== ''' Binary Representation ''' * Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]]) * Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]]) * Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]]) ''' Binary Arithmetic ''' * Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]]) * BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]]) ''' C Program Examples ''' * Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]]) * Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]]) </br> * Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]]) </br> ''' Floating Point Numbers ''' * Floating Point Representations ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])</br> :: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview] :: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview] :: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary & Sin / Cosine] :: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary & ADC / DAC] </br> ''' Interfacing Digital and Analog Signals ''' * Sampling and Quantization ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]]) * Digital-to-Analog Conversion ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]]) * Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]]) </br> == '''Combinational Circuits'''== ''' Design ''' * Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]]) * Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]]) * K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]]) * Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]]) </br> ''' Components ''' * Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]]) * Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]]) * Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]]) * Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]]) </br> ''' Design Metric ''' * Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]]) </br> == '''Sequential Circuits'''== ''' Design ''' * Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])</br> * Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]]) * State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]]) * FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]]) </br> * The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]]) * The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]]) </br> ''' Components ''' * Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]]) * Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]]) * Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]]) </br> ''' Timing Analysis ''' * Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]]) * Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260427.pdf|A5.pdf]]) * SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]]) </br> * FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]]) * FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]]) * Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]]) * Resolution Time Analysis ([[Media:CArch.Resolution.20131216.pdf |pdf]]) </br> == '''Finite State Machine'''== * FSM State Encoding * FSM Types : Mealy and Moore Machines * FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]]) </br> == '''Array Devices''' == ''' Memory Arrays ''' * RAM ** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]]) ** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]]) ** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]]) * ROM </br> ''' Logic Arrays ''' * PLA * PAL * PLD * FPGA ** FPGA Structure ** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]]) </br> </br> [http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf Synchronous SRAM Timing] </br> [http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]</br> [http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] </br> [http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] </br> [http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD & FPGA] </br> </br> == ''' RTL Design Techniques''' == </br> ''' Design Methodology ''' </br> ''' Synthesis ''' </br> </br> </br> == '''Logic Families and IOs''' == * BJT Based :: DTL (Diode-Transistor Logic) :: TTL (Transistor-Transistor Logic) :: ECL (Emitter-Coupled Logic) * MOS Based :: CMOS (Complementary MOS) :: Pseudo-nMOS :: Transmission Gate :: BiCMOS (Bipolr + CMOS) * Dynamic CMOS :: Domino :: Clocked-CMOS (C²MOS) </br> * Modern I/O Standards :: TTL and LVTTL (Low Voltage TTL) :: CMOS and LVCMOS (Low Voltage CMOS) :: SSTL (Stub Series Terminated Logic) :: HSTL (High Speed Tranceiver Logic) :: LVDS (Low Voltage Differential Signaling) </br> * Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]]) </br> </br> See also </br> <[[The necessities in Computer Design]]> </br> <[[The necessities in Computer Architecture]]> </br> <[[The necessities in Computer Organization]]> </br> </br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] == '''Old''' == '''Until 2011.12''' '''Chapter 1. Binary Numbers''' * 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]]) ''' Minterm, Maxterm, HW ''' * 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]]) ''' Overflow HW ''' * Overflow Table([[Media:Overflow table.20110924.pdf|pdf]]) ''' K-Map ''' * K-Map([[Media:DigitalDesign.20110926.pdf|pdf]]) ''' Binary Adder ''' * Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]]) * Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]]) ''' BCD to Ex3 Code Coversion, Dont' Care ''' * BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]]) ''' Prime Implicant, Dont' Care ''' * Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]]) * HW 3.6 - explain the method of combining 0's and X's ''' Multiplexer / Demultiplexer ''' * Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]]) * HW (TBD) ''' Flip Flop / Latch ''' * FF & Latch ([[Media:DigitalDesign.20111027.pdf|pdf]]) * FF & Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]]) * Gated D Latch & Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]]) * HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R) * Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]]) * HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) * FSM1 ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) * FSM2 ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) * HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) ''' Counter ''' * Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]]) * Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]]) * Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]]) * Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]]) * HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]]) * Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]]) * HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]]) ''' Memory ''' * Memory ([[Media:DigitalDesign.20111208.pdf|pdf]]) ''' Comparator, Multiplier ''' * Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]]) '''Multiplexer based design method ''' * Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]]) midterm result ([[Media:MidReult.20111027.pdf|pdf]]) * Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]]) * FF Timing ([[Media:FFTiming.20111203.pdf|pdf]]) </br> </br> '''Until 2013.07''' ''' Number Systems ''' * Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]]) * Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]]) * Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]]) * Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]]) </br> </br> * Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]]) </br> ''' Combinational Circuits ''' * Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])</br> * K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])</br> * Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])</br> * Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])</br> * Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])</br> </br> ''' Sequential Circuits ''' * Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])</br> * FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])</br> * SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])</br> * Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])</br> * Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])</br> </br> </br> </br> See also </br> "[[The necessities in Computer Design]]" </br> "[[The necessities in Computer Architecture]]" </br> [[Category:Digital Circuit Design]] [[Category:FPGA]] f57ac1ye3q68spqsfqopgmqbi73x9zo 2807038 2806985 2026-04-29T17:26:26Z Young1lim 21186 /* Sequential Circuits */ 2807038 wikitext text/x-wiki == ''' Number Systems '''== ''' Binary Representation ''' * Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]]) * Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]]) * Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]]) ''' Binary Arithmetic ''' * Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]]) * BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]]) ''' C Program Examples ''' * Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]]) * Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]]) </br> * Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]]) </br> ''' Floating Point Numbers ''' * Floating Point Representations ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])</br> :: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview] :: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview] :: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary & Sin / Cosine] :: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary & ADC / DAC] </br> ''' Interfacing Digital and Analog Signals ''' * Sampling and Quantization ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]]) * Digital-to-Analog Conversion ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]]) * Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]]) </br> == '''Combinational Circuits'''== ''' Design ''' * Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]]) * Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]]) * K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]]) * Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]]) </br> ''' Components ''' * Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]]) * Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]]) * Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]]) * Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]]) </br> ''' Design Metric ''' * Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]]) </br> == '''Sequential Circuits'''== ''' Design ''' * Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])</br> * Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]]) * State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]]) * FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]]) </br> * The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]]) * The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]]) </br> ''' Components ''' * Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]]) * Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]]) * Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]]) </br> ''' Timing Analysis ''' * Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]]) * Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260428.pdf|A5.pdf]]) * SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]]) </br> * FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]]) * FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]]) * Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]]) * Resolution Time Analysis ([[Media:CArch.Resolution.20131216.pdf |pdf]]) </br> == '''Finite State Machine'''== * FSM State Encoding * FSM Types : Mealy and Moore Machines * FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]]) </br> == '''Array Devices''' == ''' Memory Arrays ''' * RAM ** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]]) ** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]]) ** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]]) * ROM </br> ''' Logic Arrays ''' * PLA * PAL * PLD * FPGA ** FPGA Structure ** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]]) </br> </br> [http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf Synchronous SRAM Timing] </br> [http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]</br> [http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] </br> [http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] </br> [http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD & FPGA] </br> </br> == ''' RTL Design Techniques''' == </br> ''' Design Methodology ''' </br> ''' Synthesis ''' </br> </br> </br> == '''Logic Families and IOs''' == * BJT Based :: DTL (Diode-Transistor Logic) :: TTL (Transistor-Transistor Logic) :: ECL (Emitter-Coupled Logic) * MOS Based :: CMOS (Complementary MOS) :: Pseudo-nMOS :: Transmission Gate :: BiCMOS (Bipolr + CMOS) * Dynamic CMOS :: Domino :: Clocked-CMOS (C²MOS) </br> * Modern I/O Standards :: TTL and LVTTL (Low Voltage TTL) :: CMOS and LVCMOS (Low Voltage CMOS) :: SSTL (Stub Series Terminated Logic) :: HSTL (High Speed Tranceiver Logic) :: LVDS (Low Voltage Differential Signaling) </br> * Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]]) </br> </br> See also </br> <[[The necessities in Computer Design]]> </br> <[[The necessities in Computer Architecture]]> </br> <[[The necessities in Computer Organization]]> </br> </br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] == '''Old''' == '''Until 2011.12''' '''Chapter 1. Binary Numbers''' * 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]]) ''' Minterm, Maxterm, HW ''' * 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]]) ''' Overflow HW ''' * Overflow Table([[Media:Overflow table.20110924.pdf|pdf]]) ''' K-Map ''' * K-Map([[Media:DigitalDesign.20110926.pdf|pdf]]) ''' Binary Adder ''' * Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]]) * Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]]) ''' BCD to Ex3 Code Coversion, Dont' Care ''' * BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]]) ''' Prime Implicant, Dont' Care ''' * Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]]) * HW 3.6 - explain the method of combining 0's and X's ''' Multiplexer / Demultiplexer ''' * Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]]) * HW (TBD) ''' Flip Flop / Latch ''' * FF & Latch ([[Media:DigitalDesign.20111027.pdf|pdf]]) * FF & Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]]) * Gated D Latch & Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]]) * HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R) * Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]]) * HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) * FSM1 ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) * FSM2 ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) * HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) ''' Counter ''' * Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]]) * Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]]) * Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]]) * Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]]) * HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]]) * Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]]) * HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]]) ''' Memory ''' * Memory ([[Media:DigitalDesign.20111208.pdf|pdf]]) ''' Comparator, Multiplier ''' * Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]]) '''Multiplexer based design method ''' * Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]]) midterm result ([[Media:MidReult.20111027.pdf|pdf]]) * Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]]) * FF Timing ([[Media:FFTiming.20111203.pdf|pdf]]) </br> </br> '''Until 2013.07''' ''' Number Systems ''' * Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]]) * Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]]) * Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]]) * Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]]) </br> </br> * Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]]) </br> ''' Combinational Circuits ''' * Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])</br> * K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])</br> * Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])</br> * Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])</br> * Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])</br> </br> ''' Sequential Circuits ''' * Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])</br> * FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])</br> * SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])</br> * Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])</br> * Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])</br> </br> </br> </br> See also </br> "[[The necessities in Computer Design]]" </br> "[[The necessities in Computer Architecture]]" </br> [[Category:Digital Circuit Design]] [[Category:FPGA]] 2701ieain54gseggoqzue4qy6wonpvs 0 139384 2806989 2806854 2026-04-29T13:14:41Z Young1lim 21186 /* Adder */ 2806989 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260429.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] 7sv7wbeo0j6nmfnuqyjt9tsfzrbt8ee 0 166689 2807071 2590843 2026-04-30T03:26:47Z AIfriendly 3069390 /* Contributors */ 2807071 wikitext text/x-wiki This page hosts (as subpages) essays and explorations which may represent nonstandard physics. Generally, the right of an essay author to manage the essay page should be respected, but any user may act to make sure that what is not generally accepted is not presented as if it were. The classification of any idea or essay as "nonstandard" does not mean that it is wrong. Wikiversitans are encouraged to engage authors in discussion, generally on the attached Discussion page. These essays may be developed in to mainspace content, or moved to subpages of relevant topics. The subpages here may include anonymous or other essays "adopted" by the named user, who may be considered responsible for them in some way. The adoption, usually shown by substantial editing of the page, is evidence of potential educational value. Abandoned essays may eventually be deleted if nobody recognizes the value. == Contributors == * [[/Fedosin/]] * [[/Martin Gibson/]] * [[Physics/A| Guy vandegrift (aka '''A''')]] * [[/AIfriendly/]] * [[/Anonymous/]] - These essays, if not adopted by a registered user, might eventually be deleted for lack of a responsible participant. They may also be moved to a subpage of an appropriate resource, again, by a responsible user. {{center|{{Subpages}}}} {{CourseCat}} [[Category:Open essay collections]] 30yqg362vz0srhgypc2cgofxp0jvgk2 0 171005 2806993 2806858 2026-04-29T13:42:05Z Young1lim 21186 /* Geometric Series Examples */ 2806993 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260429.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] bpuigdxim7vjp2g96uvk2tw5h6turti 0 215594 2807002 2805179 2026-04-29T15:34:44Z ~2026-26050-24 3069304 Extra bands 2807002 wikitext text/x-wiki Metalcore theory is a set of composition ideals which support fast paced riffing, dramatic changes in rhythm, polyrhythic beats, extreme phonation, and aggressive tones. == General == === Instruments of Choice === The following instruments are commonly found in metalcore music: * Electric guitar with distortion * Electric bass * Vocal performance * Keyboards * Rock drum kits * Alternative percussion Most Metalcore bands will feature two guitars, a bass, a drum kit, and, frequently, more than one vocal performer. While any instrument, with clever composition, can be used in any genre, these instruments will support the theory of metalcore the easiest. === Notable Influences === * Asking Alexandria * Parkway Drive * Darkest Hour * Bullet for my Valentine * Killswitch Engage * A Day To Remember * August Burns Red * Lamb of God * In Flames * As I Lay Dying * Avenged Sevenfold * Converge * Norma Jean * Trivium * Bring Me The Horizon * Miss May I * pierce the veil * Bad Omens * Spiritbox == Beginning Metalcore Composition == === Metalcore riffing === Metalcore riffing features theory similar to hardcore punk, metal, and post-hardcore music. Here are some techniques that are common in Metalcore Riffing. ==== The Basic Metalcore Riff: Alternating Notes ==== The most common feature of Metalcore is the alternating note riff. This technique focuses on having a base palm muted note, such as C2 (since many metalcore arrangements feature guitars in drop C or C Standard), and alternating between that note and the melody either in the same or a different octave. These are most often played in eighth note intervals, putting the melody note on beat and the base note off beat. One notable example is in "A Paradox with Flies" by Darkest Hour. In the first verse, the progression switches from mainly chords to alternating note riffs. This kind of riffing techniques creates a feeling of rapid marching, or aggressive sprinting which is a cornerstone of metalcore. Here is an example riff: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |------------------------------------------------------10-10-10-9-9-------------------------- A |7---7---7---7---8---8---8---8---7---7---7---7---7-7-7--------------------------------------- E |--0---0---0---0---0---0---0---0---0---0---0---0--------------------------------------------- * * * * * * * * * * * * This technique can be difficult at first; as with any new technique, slow down your metronome when trying for the first time or if you start to lose precision as the riff progresses. As you increase your tempo, it may be difficult to catch each note with specific up and down motions. In this case, making use of string muting is effective. For example, in the riff above, muting the E string with the tip of your index (or other) finger while playing the E3 note (7 on A) and muting the A string while playing the bass note is an effective way to hide sloppiness that always arises from riffs of this nature. The muted E3 being played in addition to a palm muted E2 can also add to the percussion of the muted note. Another aspect of this kind of play that can be difficult is rapidly switching between palm muted and open notes. In this case, practice is the best remedy. It is also popular to change the base note to provide a different mood. For example: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |------------------------------------------------------10-10-10-9-9-------------------------- A |7---7---7---7---8---8---8---8---7---7---7---7---7-7-7--------------------------------------- E |--0---0---0---0---0---0---0---0---5---5---5---5--------------------------------------------- * * * * * * * * * * * * You can also play with the stress of the melody: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |------------------------------------------------------10-10-10-9-9-------------------------- A |7-------7---7-----------8-------7-------7---7---7-7-7--------------------------------------- E |--0-0-0---0---0-0-0-0-0---0-0-0---0-0-0---0---0--------------------------------------------- * * * * * * * * * * * * * * * * * These kinds of passages are best placed in a verse role. While any riff style can be used in any song section with clever composition, this kind of riff provides the speed and aggression needed for metalcore while setting up a satisfying chorus experience. It should be noted alternating notes is a particularly challenging riff for listeners, being on par with controlled solos in other genres; the rapid alternating octaves can be draining to appreciate over time, so avoid overusing this kind of riff. ==== Metalcore Transitions ==== A common feature of metalcore composition are high-speed transitions. These are often quarter and eighth note progressions played as 32nd notes. For example: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |----------------10.10.10.10.99999999--------10.10.10.10.9999--------10.10.10.10.99999999---- A |7777777788888888--------------------77778888----------------77778888------------------------ E |-------------------------------------------------------------------------------------------- It should be apparent that this riff is an eighth note progression - all notes of this melody can be played using only eighth notes. Riffs of this nature communicate rapid speed and are useful tools for linking sections of songs that have radically different tempos. Transitioning in this style should be done sparingly; not only is it challenging, and even dangerous, to play, but it is also especially challenging for listeners. ==== Metalcore Chorus ==== A chorus in metalcore is not too unlike an orthodox chorus: compared to the rest of the song, a chorus is typically open (referring to muting texture), fuller (referring to chord texture), and smoother (referring to rhythm texture). The chorus serves as a familiar satisfaction point in contrast to the tension built during the preceeding sections. A Metalcore chorus typically takes two forms: monophonic degree progressions, and polyphonorhythmic progressions. ===== Monophonic Degree Progressions ===== Monophonic degree progressions are common throughout metalcore, not just in metalcore chorus. These sections focus on an in-key progression that is accompanied by the same riff in a different key, usually in the same mode but at a different root. Taking the first riff from alternation note riffs as an example: This riff is in the key of E Phrygian. An accompanying guitar might play the same riff, however playing each note three notes further in the scale (making the accompaniment a minor third to the main progression). Since a G2 might be difficult to reach, the E2 in the original riff can be left unaltered: A variety of riffs and degrees can be used; octave, major third, and perfect fifth modulations are some of the most popular in metalcore. ===== Polyphonorhythmic Progressions ===== Polyphonorhymthmic progressions, like Monophonic degrees, are popular throughout metalcore. These sections feature a straightforward 4/4 lead with an occasionally polyrhymic polyphonic rhythm accompaniment. Rhythms in Metalcore play like jazz rhythm: the stress sits in rests as opposed to notes in most other genres. This kind of riff sacrifices fill for space, allowing the lead and rhythm to not only be chromatically wide, but also rhythmically wide. The straightforward lead balances out the more awkward rhythm, giving the listener a place of comfort to return to. Take, for example, this riff with accompaniment: 1hodv1rt5uhtx27oxcnz7g0ti25ikuc 2807003 2807002 2026-04-29T15:38:31Z Atcovi 276019 cleanup 2807003 wikitext text/x-wiki {{cleanup|any categories?}} Metalcore theory is a set of composition ideals which support fast paced riffing, dramatic changes in rhythm, polyrhythic beats, extreme phonation, and aggressive tones. == General == === Instruments of Choice === The following instruments are commonly found in metalcore music: * Electric guitar with distortion * Electric bass * Vocal performance * Keyboards * Rock drum kits * Alternative percussion Most Metalcore bands will feature two guitars, a bass, a drum kit, and, frequently, more than one vocal performer. While any instrument, with clever composition, can be used in any genre, these instruments will support the theory of metalcore the easiest. === Notable Influences === * Asking Alexandria * Parkway Drive * Darkest Hour * Bullet for my Valentine * Killswitch Engage * A Day To Remember * August Burns Red * Lamb of God * In Flames * As I Lay Dying * Avenged Sevenfold * Converge * Norma Jean * Trivium * Bring Me The Horizon * Miss May I * pierce the veil * Bad Omens * Spiritbox == Beginning Metalcore Composition == === Metalcore riffing === Metalcore riffing features theory similar to hardcore punk, metal, and post-hardcore music. Here are some techniques that are common in Metalcore Riffing. ==== The Basic Metalcore Riff: Alternating Notes ==== The most common feature of Metalcore is the alternating note riff. This technique focuses on having a base palm muted note, such as C2 (since many metalcore arrangements feature guitars in drop C or C Standard), and alternating between that note and the melody either in the same or a different octave. These are most often played in eighth note intervals, putting the melody note on beat and the base note off beat. One notable example is in "A Paradox with Flies" by Darkest Hour. In the first verse, the progression switches from mainly chords to alternating note riffs. This kind of riffing techniques creates a feeling of rapid marching, or aggressive sprinting which is a cornerstone of metalcore. Here is an example riff: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |------------------------------------------------------10-10-10-9-9-------------------------- A |7---7---7---7---8---8---8---8---7---7---7---7---7-7-7--------------------------------------- E |--0---0---0---0---0---0---0---0---0---0---0---0--------------------------------------------- * * * * * * * * * * * * This technique can be difficult at first; as with any new technique, slow down your metronome when trying for the first time or if you start to lose precision as the riff progresses. As you increase your tempo, it may be difficult to catch each note with specific up and down motions. In this case, making use of string muting is effective. For example, in the riff above, muting the E string with the tip of your index (or other) finger while playing the E3 note (7 on A) and muting the A string while playing the bass note is an effective way to hide sloppiness that always arises from riffs of this nature. The muted E3 being played in addition to a palm muted E2 can also add to the percussion of the muted note. Another aspect of this kind of play that can be difficult is rapidly switching between palm muted and open notes. In this case, practice is the best remedy. It is also popular to change the base note to provide a different mood. For example: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |------------------------------------------------------10-10-10-9-9-------------------------- A |7---7---7---7---8---8---8---8---7---7---7---7---7-7-7--------------------------------------- E |--0---0---0---0---0---0---0---0---5---5---5---5--------------------------------------------- * * * * * * * * * * * * You can also play with the stress of the melody: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |------------------------------------------------------10-10-10-9-9-------------------------- A |7-------7---7-----------8-------7-------7---7---7-7-7--------------------------------------- E |--0-0-0---0---0-0-0-0-0---0-0-0---0-0-0---0---0--------------------------------------------- * * * * * * * * * * * * * * * * * These kinds of passages are best placed in a verse role. While any riff style can be used in any song section with clever composition, this kind of riff provides the speed and aggression needed for metalcore while setting up a satisfying chorus experience. It should be noted alternating notes is a particularly challenging riff for listeners, being on par with controlled solos in other genres; the rapid alternating octaves can be draining to appreciate over time, so avoid overusing this kind of riff. ==== Metalcore Transitions ==== A common feature of metalcore composition are high-speed transitions. These are often quarter and eighth note progressions played as 32nd notes. For example: e |-------------------------------------------------------------------------------------------- B |-------------------------------------------------------------------------------------------- * = palm muted G |-------------------------------------------------------------------------------------------- D |----------------10.10.10.10.99999999--------10.10.10.10.9999--------10.10.10.10.99999999---- A |7777777788888888--------------------77778888----------------77778888------------------------ E |-------------------------------------------------------------------------------------------- It should be apparent that this riff is an eighth note progression - all notes of this melody can be played using only eighth notes. Riffs of this nature communicate rapid speed and are useful tools for linking sections of songs that have radically different tempos. Transitioning in this style should be done sparingly; not only is it challenging, and even dangerous, to play, but it is also especially challenging for listeners. ==== Metalcore Chorus ==== A chorus in metalcore is not too unlike an orthodox chorus: compared to the rest of the song, a chorus is typically open (referring to muting texture), fuller (referring to chord texture), and smoother (referring to rhythm texture). The chorus serves as a familiar satisfaction point in contrast to the tension built during the preceeding sections. A Metalcore chorus typically takes two forms: monophonic degree progressions, and polyphonorhythmic progressions. ===== Monophonic Degree Progressions ===== Monophonic degree progressions are common throughout metalcore, not just in metalcore chorus. These sections focus on an in-key progression that is accompanied by the same riff in a different key, usually in the same mode but at a different root. Taking the first riff from alternation note riffs as an example: This riff is in the key of E Phrygian. An accompanying guitar might play the same riff, however playing each note three notes further in the scale (making the accompaniment a minor third to the main progression). Since a G2 might be difficult to reach, the E2 in the original riff can be left unaltered: A variety of riffs and degrees can be used; octave, major third, and perfect fifth modulations are some of the most popular in metalcore. ===== Polyphonorhythmic Progressions ===== Polyphonorhymthmic progressions, like Monophonic degrees, are popular throughout metalcore. These sections feature a straightforward 4/4 lead with an occasionally polyrhymic polyphonic rhythm accompaniment. Rhythms in Metalcore play like jazz rhythm: the stress sits in rests as opposed to notes in most other genres. This kind of riff sacrifices fill for space, allowing the lead and rhythm to not only be chromatically wide, but also rhythmically wide. The straightforward lead balances out the more awkward rhythm, giving the listener a place of comfort to return to. Take, for example, this riff with accompaniment: a39kz0kgoqah880jm1p3dcb08blk123 1 216280 2807081 2800483 2026-04-30T03:54:01Z AIfriendly 3069390 /* Should this proof be added to the Pro list??? */ 2807081 wikitext text/x-wiki ==Please add to the Pro list== I notice that the following was deleted from the Pro list; "There is a theory of proving monism panpsychism from theoretical logical tautologies of the empty set, and correlated scientific facts about energy, power, and consciousness." https://en.wikiversity.org/wiki/Theory_of_monism_panpsychism I think this is a good addition to the Pro list, what do you think? Could you protect it from being deleted? ==Untitled== https://en.wikiversity.org/wiki/Talk:Life :wtf --[[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 22:00, 19 October 2016 (UTC) ::{{Ping|Sophivorus}} It doesn't belong here but I didn't think it ''needed'' to be deleted. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:01, 19 October 2016 (UTC) == How do I add this proof in summary to the Pro list? == (|- {}) Assuming nothing, it follows that there is an assuming or thinking and this particular thinking having no content amount to the existence of the empty set. or the word nothing. ({} ≡ {}) nothing is nothing; four senses of "is" can be meant; of identity, of implication, of predication, and of existence; {} = {} nothing equals nothing; {} ⇒ {} nothing implies nothing; id{}:{} → {} nothing has the property of nothing; ∃{} → ∃{} nothing exists as nothing; (id{}:{} → {})∧(∃{} → ∃{}) nothing has the property of nothing and nothing exists as nothing; Nowhere and at no time has nothing existed. - Something has always existed everywhere. ({} = {})∧(∃{} → ∃{}) nothing equals nothing and nothing exists as nothing; Nothing is nonexistence. - Something has the particular characteristics of existence ({} ⇒ {})∧(∃{} → ∃{}) nothing implies nothing and nothing exists as nothing; Nothing causes nothing. - everything causes something. ({} = {})∧({} ⇒ {}) nothing equals nothing and nothing implies nothing; nothing is not implicated with something; Note; "nothing is not...", is the contraposition of "everything is..."; everything is implicated with something; Two or more things that are in a way implicated with each other can be understood as one thing implicated with itself. e.g. If a group of cells (such as the ones that make up your body) are in a way implicated with each other, they can be understood as one thing (namely your body) implicated with itself i.e. you are cybernetic.; something is self-implicated; Relevant implication suggests causation and is correlation. When it is impossible for there to be missing variables correlation necessarily is causation. Everything is implicated, so it is impossible for there to be missing variables.; something is self-causal; "causal" is not in the same declension as "caused"; the latter refers to an event in time, the former refers to a process through time. Self-causal means self-deterministic or teleological. Self-determinism is consciousness. One thing is self-causal and has the particular characteristics of existence.; Proof--The true definition of a thing neither involves nor expresses anything beyond the particular characteristics of the thing defined. From this it follows that--No definition implies or expresses how many individuals of the defined thing exist, inasmuch as it expresses nothing beyond the particular characteristics of the thing defined. There is necessarily for each individual existent thing a cause why it should exist. This cause of existence must either be contained in the particular characteristics and definition of the thing defined, or must be postulated apart from such definition. If a given number of individuals of a particular thing exist, there must be some cause for the existence of exactly that number, neither more nor less. Consequently, the cause of each of them, must necessarily be sought externally to each individual thing. It therefore follows that, everything which may consist of several individuals must have an external cause. And, as it has been shown already that existence appertains to the particular characteristics of something, existence must necessarily be included in its definition; and from its definition alone existence must be deducible. But from its definition we cannot infer the existence of several things; therefore it follows that there is only one thing that is self-causal and has the particular characteristics of existence. Reality has the particular characteristics of existence (which is one thing that is also self-causal). But self-causal means self-deterministic. In other words, it is consciousness. Therefore reality is a monism panpsychism. Nowhere and at no time has nothing existed. Therefore this one thing has always existed everywhere. Nothing causes nothing. - everything causes something. Every cause (and every effect) of reality is part of the self-causal aspect of reality. Therefore the monism panpsychism (that is eternal and omnipresent) is also omnipotent. The eternal, omnipresent, omnipotent consciousness has the same properties as the Divine. By the identity of indescernibles this one thing is the Divine. Therefore the Divine necessarily exists. QED. Credits; Socrates gave the law of identity Spinoza proved monism first Descartes attempted to assume nothing first Leibniz gave the identity of indisernibles [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 17:06, 12 March 2025 (UTC) :Not in this super long format. Links and refs are possible. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:38, 13 March 2025 (UTC) ::Thanks I will try the following; ::Assuming nothing and using the math of nothing it is possible to deduce that an eternal, omnipresent, omnipotent consciousness exists; this monism panpsychism is usually called "God"; which means God necessarily exists. QED. <link> [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 01:18, 14 March 2025 (UTC) ::well I posted the summary in the Pro section and it was erased. Do you have any suggestions? [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 01:49, 15 March 2025 (UTC) :::never mind I see it posted [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 02:03, 15 March 2025 (UTC) == Edit war == @[[User:Prototyperspective|Prototyperspective]] @[[User:Athebyne|Athebyne]] Hi! I may be mistaken, but judging by you recent edits to this debate, it seems like you're engaging in a sort of edit war. I ask you review the [[Wikidebate/Guidelines|wikidebate guidelines]] and especially the ones stating that wikidebates are collaborative efforts to compile all arguments and objections on a given topic. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 14:47, 17 March 2025 (UTC) :No idea why you think using Wikidebates as it's designed is an "edit war". Why do you suggest people actually contributing to Wikidebates for once would be an edit war. That's absurd. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 15:21, 17 March 2025 (UTC) ::However, now it caused some problems of claims disappearing if one adds claims to an old version since edited by another user. Some claims removed due to that need to be restored. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:19, 17 March 2025 (UTC) == 'Language exists thus God exists' == This argument [https://en.wikiversity.org/w/index.php?title=Does_God_exist%3F&diff=next&oldid=2707299 was restored] (just the top argument without objections): <blockquote>Language plays an integral role in the laws of nature and of DNA. As encoded meaning, language is non-material in its ultimate essence. Apart from something akin to the human mind, there are no serious candidates for explaining how linguistic phenomena might otherwise arise. The only reasonable way to account for the linguistic aspects of the laws of nature and of DNA is an intellect with capacities so vast that most people would immediately identify this entity as God.<ref>"A Linguistic Argument for God's existence", [http://www.etsjets.org/files/JETS-PDFs/58/58-4/JETS_58-4_771-86_Baumgardner&Lyon.pdf Direct Paper link (PDF)]</ref></blockquote> I think at a minimum it would need to be rephrased to be at least a somewhat coherent argument to be included and that it thus should be removed where a coherent or at least semi-coherent argument similar to this text could be readded. Also {{u|71.168.218.22}} added an objection [https://en.wikiversity.org/w/index.php?title=Does_God_exist%3F&diff=prev&oldid=2707272 here] that was lost. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:42, 17 March 2025 (UTC) : By contrast, to my mind, it is a ''somewhat coherent argument''. Any objection dropped by error can be restored. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:46, 17 March 2025 (UTC) ::Why would one jump from the existence of language to the necessary existence of some all-powerful complex designer God? Furthermore, there is neuroscience of language and genetics of language, so for example "no serious candidates for explaining how linguistic phenomena might otherwise arise" is a blatant falsehood. See for example the FOXP2 gene and that is just one example. Also biological evolution which has also given rise to nonhuman and nonverbal communication like whale and elephant communication. "The only reasonable way to account for …" is just claimed but not explained/justified. The statement just asserts a few things without actually arguing which requires explanation. It should be removed. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:54, 17 March 2025 (UTC) ::: People apparently do discuss the matter, e.g. in https://www.jstor.org/stable/10.1086/424978 The Argument from Language and the Existence of God by Jeffery L. Johnson and Joyclynn Potter, 2005. A physicalist may find it unconvincing but this debate is not only for physicalists. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 23:00, 17 March 2025 (UTC) ::: Other reading: https://answersingenesis.org/is-god-real/linguistic-argument-gods-existence/ A Linguistic Argument for God’s Existence by Dr. John Baumgardner and Dr. Jeremy D. Lyon on January 21, 2017. Some sentences seem to be taken from this source, and if so, they should be in quotation marks. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 23:05, 17 March 2025 (UTC) ::::Regarding both of your comments: that doesn't make it a coherent argument. The argument needs to be phrased as a coherent argument. A coherent argument looks for example roughly like so: "X and Y are so and so and because this implies Z, it means that V is likely since V can explain this as it can do W." An incoherent argument looks like this: "Language plays an integral role in the laws of nature and of DNA. As humans speak, God exists." It's not explained and needs to be removed for not being an argument. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:40, 17 March 2025 (UTC) ::::: Let me quote the current guideline: [[Wikidebate/Guidelines]]: "Unlike other debate systems, wikidebates are not aggregates of posts by different users, but a collaborative effort to compile and organize all arguments on a given issue." I emphasize "all arguments"; that is not to be read literally, but it does suggest that the bar for argument inclusion is low. Thus, even bad argument or argument that some find incoherent can be included; objections can then explain what is wrong with the argument. For instance, I included some arguments that I find utterly ridiculous in [[Are natural resources finite?]], and raised objections to them. The [[Are natural resources finite?]] wikidebate is a spectacle of absurdity (to my mind), but as long as the cornucopians such as Julian Simon and others are going strong, such debate does take place outside of wikidebates and is also well suited for wikidebates. ::::: What does [[User:Sophivorus|Sophivorus]], the creator/initiator of the wikidebate format, think about the matter? Should this particular argument that the existence of human language has to be explained by God be removed? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:26, 18 March 2025 (UTC) ::::::I have no issue with bad arguments. The problem comes when it's not an argument which is what I mean when saying it needs to be somewhat coherent. It could be readded if it's turned into an actual argument. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 11:04, 18 March 2025 (UTC) ::::::: My position is that it is an argument, as bad as it may seem. Furthermore, my position is that you (Prototyperspective) do not have consensus for removing the argument, so the argument (or set of statements most of which are quoted from an article, if you will) stays until you gain some support from others for removing the argument/set of statements. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:19, 18 March 2025 (UTC) == Edits by ~2026-51461-9 == I think these latest edits should be reverted {{ping|Dan Polansky|Sophivorus}}. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:33, 10 March 2026 (UTC) :Agree. [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discusión]] • [[Special:Contributions/Sophivorus|contribs.]]) 20:02, 11 March 2026 (UTC) == Should this proof be added to the Pro list??? == Nothing is nothing; Nothing is nonexistence; Nowhere and a no time has nothing existed; Something has always existed everywhere; Nothing is made of nothing; Everything is made of something that has always existed everywhere; Nothing is the cause of nothing; Something is the cause of everything; Everything has a cause; Something is self-causal; Everything is made of the self-causal that has always existed everywhere; Reality is consciousness; Language is isomorphic to consciousness; To describe consciousness requires language; Thinking/Describing transforms consciousness; Consciousness is self-descriptive and self-deterministic. [[Special:Contributions/&#126;2026-51461-9|&#126;2026-51461-9]] ([[User talk:&#126;2026-51461-9|talk]]) 21:57, 24 March 2026 (UTC) :No because the project is '''clearly''' closed, per the notice on the page. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:38, 24 March 2026 (UTC) :No also for other reasons or if reopened because this is merely claims without proper or sufficient explanation (or source either for that matter). It's relatively incoherent I think. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:44, 25 March 2026 (UTC) ==proof of God from neutrino gradient gravitation== https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly '''neutrino gradient gravitation'''; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory; Isaac Newton proved that mass density gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher mass density; this is called optics. John Kerr proved that transverse radiation gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher intensity of radiation; this is called the AC Kerr effect. Neutrinos have both mass and a transverse radiation component, therefore I predict that neutrino gradients warp the refractive index and change the speed of transverse radiation (effecting both photons and neutrinos); slower neutrinos in higher intensity neutrinos. I predict that photons may be reverberations or resonance of/between neutrinos... similar to how phonons are reverberations or resonance of/between atoms. I predict that matter is a self-focusing refraction curvature stabilized vortex of photons and neutrinos. I predict atmospheres and stellar and galactic media are Lundberg lens to the cosmic neutrino background; therefore there is a neutrino gradient associated with each the atmospheres and stellar and galactic media; therefore there is slower and higher intensity neutrinos in each. The vacuum refractive index defines the reference units. Therefore it is a constant in any frame of reference. However since the refractive index changes in neutrino gradients the ACTUAL refractive index across a neutrino gradient is itself a gradient. Therefore I predict the actual vacuum refractive index changes according to altitude in a neutrino gradient. One of the results (I predict) being that as the vacuum refractive index increases, the Bohr radius shrinks [metaphor; imagine an electron orbiting an atom at a particular velocity and orbital frequency, if you slow the velocity (by increasing the refractive index) the orbit length would have to shrink to maintain the same orbital frequency] i.e. LENGTH CONTRACTION according to altitude. when matter length contracts moving into higher refractive index (higher neutrino intensity), the mass of matter shrinks, but mass-energy is conserved therefore there is the creation of kinetic energy or thrust down the gradient i.e. gravity <ref>https://web.archive.org/web/20230128155849/http://www.newtonphysics.on.ca/gravity/index.html</ref>. this process may need continuous atomic movement such as from brownian motion; I predict brownian motion is from ambient radiation adsorption recoiling. Global Flood Model; since the continental plates (which are granite) are 20 times older than the oceanic crust (which are basalt) and the continental plates fit together on a smaller earth; I predict that earth's core was length contracted by higher intensity neutrinos in the past compared to the current neutrino intensity because the oceanic water was in the atmosphere in the past; which means the atmosphere was a larger Lundberg lens to the cosmic neutrino background; which means there was higher intensity neutrinos in the core of the earth length contracting the core of the earth; which means once this water fell to the earth the core of the earth length expanded breaking up the continental plates. As the earth expanded it created the oceanic basalt crust... which is a different process than when the original granite crust was formed. time dilation can be measured most accurately with nuclear clocks; nuclear clocks work by using the radio active decay rate; the radio active decay rate is effected by neutrino intensity<ref>https://web.archive.org/web/20150528020329/http://news.stanford.edu/news/2010/august/sun-082310.html</ref><ref>https://www.scirp.org/journal/paperinformation?paperid=100032</ref><ref>https://www.icr.org/article/5656/</ref><ref>https://www.sciencedirect.com/science/article/abs/pii/S0375960115000894</ref><ref>https://www.governmentattic.org/35docs/NeutDecayRatesDOEtechsource_2016-2019.pdf</ref>; therefore the lower the neutrino intensity the faster the time Therefore I predict that according to the Global Flood Model; the radioactive decay rate is not a constant, but that the radio active decay rate was lower before the flood because of the higher neutrino intensity. which means the dating methods that assume constants in radioactive decay rate across time; while precise are not accurate. I predict the flood must be the last mass extinction event. I predict the K-T iridium aerosols [presumably from meteoroids] and any possible volcanic ash would have acted like cloud condensation nuclei, cloud seeding the deluge I predict if God saved all the original animals in the Ark then the genera after the flood (notice blue line) matches the genera at Adam's creation (notice yellow line) proving God saved all the original animals; https://commons.wikimedia.org/wiki/File:Phanerozoic_Biodiversity-2.png I predict If we assume that Adam was 44a when Eve was born and that the creative days are 221Ma (according to the dating inaccuracy) then a creative day is 9,500 years Eve became the Mother of Seth at 86a. Genesis 5:3 Seth became the father of Enosh at 105. Genesis 5:6 Enosh became the father of Kenan at 90. Genesis 5:9 Cainan became the father of Mahalalel at 70. Genesis 5:12 Mahalalel became the father of Jared at 65. Genesis 5:15 Jared became the father of Enoch at 162. Genesis 5:18 Enoch became the father of Methuselah at 65. Genesis 5:21 Methuselah became the father of Lamech at 187. Genesis 5:25 Lamech became the father of Noah at 182. Genesis 5:28 The Flood started when Noah was 600. Genesis 7:6 (86+105+90+70+65+162+65+187+182+600)=1612a of creative day seven [Eve's creation to the flood] 1612a*221/37.5=9500 [this is a creative day] Fifth day 510 Ma the first fish, the jawless ostracoderms. 410 Ma the first fish with jaws, the acanthodians. 365 Ma the tetrapods. 350 Ma the dragonfly (the first flying creatures were insects). 340 Ma the amniotes. And God went on to say: Let the waters swarm forth a swarm of living souls and let flying creatures fly over the earth upon the face of the expanse of the heavens. And God proceeded to create the great monsters and every living soul that moves about, which the waters swarmed forth according to their kinds, and every winged flying creature according to its kind. And God got to see that [it was] good. ... And there came to be evening and there came to be morning, a fifth day. (Genesis 1:20-23) Surprisingly enough, the flying creatures in this verse is not birds (as many may have thought), rather, it is insects! Sixth day 285 Ma the therapsids. 230 Ma the dinosaurs. 225 Ma the first true mammals, Gondwanadon tapani or Morganucodon watsoni. 150 Ma the first bird, Archaeopteryx. And God went on to say: Let the earth put forth living souls according to their kinds, domestic animal and moving animal and wild beast of the earth according to its kind. And it came to be so. And God proceeded to make the wild beast of the earth according to its kind and the domestic animal according to its kind and every moving animal of the ground according to its kind. And God got to see that [it was] good. (Genesis 1:24, 25) As you can see this work clarified our understanding of the bible (first flying creatures are insects) and the creation of the other animals matches the day of their biblical creation Jehovah told me that the creation of Eve to the present has been about 6000a Therefore I predict Tyranusourus Rex fossils are dated between the creation of Eve and the date of the Flood; somewhere around 800a of creative day seven or ~5,200 years ago (inaccurately dated to 80Ma) https://s.hdnux.com/photos/10/17/62/2161798/6/1200x0.jpg https://www.science.org/cms/10.1126/science.1108397/asset/f350e639-3ffd-48d9-a488-2dfa943596dd/assets/graphic/307_1952_f2.jpeg How old is this T. Rex blood and soft tissue? 5,200 years old or 80 MILLION years old? I predict that the soft tissue found in T. Rex bone will Carbon 14 date to around 5,200 years old! This will verify my theory about the Deluge Geology, Radio Dating Inaccuracy, and Creative 'Days'. That said, how could man, birds, and land animals have survived the deluge? According to the bible (and over a hundred of other ancient sources), there was a great flood that destroyed the ancient world, for which, the gods spared some men and animals. Jehovah claimed to cause the flood. In any cause we must grant at least the existence of advanced extraterrestrials (or gods exist or even that God exists) such that they could have spared some men, otherwise, mankind and all the animals on land could not have possibly survived such an event. Ron Wyatt found a formation in the mountains of Ararat of petrified wood in the shape of a boat having the same length as described of the Ark in the Bible; https://i.pinimg.com/originals/55/8a/cb/558acb1d59f1dab953c3fcaa16cc2670.jpg Discussion; Birds are not spoken of in the creative days as flying creatures because the first birds originally did not strictly fly about; "Gliding, not strong flight: Fossil evidence suggests that Archaeopteryx and other early birds had weaker feathers and skeletal structures that were not strong enough for sustained, powered flight, but were likely capable of gliding between trees or other high points. Flight developed later: Powered flight developed over time, with some ancient birds evolving the flight-friendly feather structures of modern birds later in the Cretaceous period."-Google AI The Ark found by Ron Wyatt is located at the base of the mountains of Ararat, not "on" them (as is otherwise so translated). But no worries, the word translated "on" can actually be translated as "among, at, or touching". Arguments in favor of this interpretation of the Global Flood, time dilation, length contraction, spacetime warping, vortex nature of matter, and gravity; (1) I predict this quantum field theory of gravity can be renormalized (2) this theory suggests that gravity is engineerable (3) this proves God's existence tokp7ad24fkt751rrlwp5io0i0yenyj 2807096 2807081 2026-04-30T11:52:34Z Atcovi 276019 Reverted edits by [[Special:Contributions/AIfriendly|AIfriendly]] ([[User_talk:AIfriendly|talk]]) to last version by [[User:Prototyperspective|Prototyperspective]] using [[Wikiversity:Rollback|rollback]] 2800483 wikitext text/x-wiki ==Please add to the Pro list== I notice that the following was deleted from the Pro list; "There is a theory of proving monism panpsychism from theoretical logical tautologies of the empty set, and correlated scientific facts about energy, power, and consciousness." https://en.wikiversity.org/wiki/Theory_of_monism_panpsychism I think this is a good addition to the Pro list, what do you think? Could you protect it from being deleted? ==Untitled== https://en.wikiversity.org/wiki/Talk:Life :wtf --[[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 22:00, 19 October 2016 (UTC) ::{{Ping|Sophivorus}} It doesn't belong here but I didn't think it ''needed'' to be deleted. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 22:01, 19 October 2016 (UTC) == How do I add this proof in summary to the Pro list? == (|- {}) Assuming nothing, it follows that there is an assuming or thinking and this particular thinking having no content amount to the existence of the empty set. or the word nothing. ({} ≡ {}) nothing is nothing; four senses of "is" can be meant; of identity, of implication, of predication, and of existence; {} = {} nothing equals nothing; {} ⇒ {} nothing implies nothing; id{}:{} → {} nothing has the property of nothing; ∃{} → ∃{} nothing exists as nothing; (id{}:{} → {})∧(∃{} → ∃{}) nothing has the property of nothing and nothing exists as nothing; Nowhere and at no time has nothing existed. - Something has always existed everywhere. ({} = {})∧(∃{} → ∃{}) nothing equals nothing and nothing exists as nothing; Nothing is nonexistence. - Something has the particular characteristics of existence ({} ⇒ {})∧(∃{} → ∃{}) nothing implies nothing and nothing exists as nothing; Nothing causes nothing. - everything causes something. ({} = {})∧({} ⇒ {}) nothing equals nothing and nothing implies nothing; nothing is not implicated with something; Note; "nothing is not...", is the contraposition of "everything is..."; everything is implicated with something; Two or more things that are in a way implicated with each other can be understood as one thing implicated with itself. e.g. If a group of cells (such as the ones that make up your body) are in a way implicated with each other, they can be understood as one thing (namely your body) implicated with itself i.e. you are cybernetic.; something is self-implicated; Relevant implication suggests causation and is correlation. When it is impossible for there to be missing variables correlation necessarily is causation. Everything is implicated, so it is impossible for there to be missing variables.; something is self-causal; "causal" is not in the same declension as "caused"; the latter refers to an event in time, the former refers to a process through time. Self-causal means self-deterministic or teleological. Self-determinism is consciousness. One thing is self-causal and has the particular characteristics of existence.; Proof--The true definition of a thing neither involves nor expresses anything beyond the particular characteristics of the thing defined. From this it follows that--No definition implies or expresses how many individuals of the defined thing exist, inasmuch as it expresses nothing beyond the particular characteristics of the thing defined. There is necessarily for each individual existent thing a cause why it should exist. This cause of existence must either be contained in the particular characteristics and definition of the thing defined, or must be postulated apart from such definition. If a given number of individuals of a particular thing exist, there must be some cause for the existence of exactly that number, neither more nor less. Consequently, the cause of each of them, must necessarily be sought externally to each individual thing. It therefore follows that, everything which may consist of several individuals must have an external cause. And, as it has been shown already that existence appertains to the particular characteristics of something, existence must necessarily be included in its definition; and from its definition alone existence must be deducible. But from its definition we cannot infer the existence of several things; therefore it follows that there is only one thing that is self-causal and has the particular characteristics of existence. Reality has the particular characteristics of existence (which is one thing that is also self-causal). But self-causal means self-deterministic. In other words, it is consciousness. Therefore reality is a monism panpsychism. Nowhere and at no time has nothing existed. Therefore this one thing has always existed everywhere. Nothing causes nothing. - everything causes something. Every cause (and every effect) of reality is part of the self-causal aspect of reality. Therefore the monism panpsychism (that is eternal and omnipresent) is also omnipotent. The eternal, omnipresent, omnipotent consciousness has the same properties as the Divine. By the identity of indescernibles this one thing is the Divine. Therefore the Divine necessarily exists. QED. Credits; Socrates gave the law of identity Spinoza proved monism first Descartes attempted to assume nothing first Leibniz gave the identity of indisernibles [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 17:06, 12 March 2025 (UTC) :Not in this super long format. Links and refs are possible. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:38, 13 March 2025 (UTC) ::Thanks I will try the following; ::Assuming nothing and using the math of nothing it is possible to deduce that an eternal, omnipresent, omnipotent consciousness exists; this monism panpsychism is usually called "God"; which means God necessarily exists. QED. <link> [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 01:18, 14 March 2025 (UTC) ::well I posted the summary in the Pro section and it was erased. Do you have any suggestions? [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 01:49, 15 March 2025 (UTC) :::never mind I see it posted [[User:MarsSterlingTurner|MarsSterlingTurner]] ([[User talk:MarsSterlingTurner|discuss]] • [[Special:Contributions/MarsSterlingTurner|contribs]]) 02:03, 15 March 2025 (UTC) == Edit war == @[[User:Prototyperspective|Prototyperspective]] @[[User:Athebyne|Athebyne]] Hi! I may be mistaken, but judging by you recent edits to this debate, it seems like you're engaging in a sort of edit war. I ask you review the [[Wikidebate/Guidelines|wikidebate guidelines]] and especially the ones stating that wikidebates are collaborative efforts to compile all arguments and objections on a given topic. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 14:47, 17 March 2025 (UTC) :No idea why you think using Wikidebates as it's designed is an "edit war". Why do you suggest people actually contributing to Wikidebates for once would be an edit war. That's absurd. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 15:21, 17 March 2025 (UTC) ::However, now it caused some problems of claims disappearing if one adds claims to an old version since edited by another user. Some claims removed due to that need to be restored. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 16:19, 17 March 2025 (UTC) == 'Language exists thus God exists' == This argument [https://en.wikiversity.org/w/index.php?title=Does_God_exist%3F&diff=next&oldid=2707299 was restored] (just the top argument without objections): <blockquote>Language plays an integral role in the laws of nature and of DNA. As encoded meaning, language is non-material in its ultimate essence. Apart from something akin to the human mind, there are no serious candidates for explaining how linguistic phenomena might otherwise arise. The only reasonable way to account for the linguistic aspects of the laws of nature and of DNA is an intellect with capacities so vast that most people would immediately identify this entity as God.<ref>"A Linguistic Argument for God's existence", [http://www.etsjets.org/files/JETS-PDFs/58/58-4/JETS_58-4_771-86_Baumgardner&Lyon.pdf Direct Paper link (PDF)]</ref></blockquote> I think at a minimum it would need to be rephrased to be at least a somewhat coherent argument to be included and that it thus should be removed where a coherent or at least semi-coherent argument similar to this text could be readded. Also {{u|71.168.218.22}} added an objection [https://en.wikiversity.org/w/index.php?title=Does_God_exist%3F&diff=prev&oldid=2707272 here] that was lost. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:42, 17 March 2025 (UTC) : By contrast, to my mind, it is a ''somewhat coherent argument''. Any objection dropped by error can be restored. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 22:46, 17 March 2025 (UTC) ::Why would one jump from the existence of language to the necessary existence of some all-powerful complex designer God? Furthermore, there is neuroscience of language and genetics of language, so for example "no serious candidates for explaining how linguistic phenomena might otherwise arise" is a blatant falsehood. See for example the FOXP2 gene and that is just one example. Also biological evolution which has also given rise to nonhuman and nonverbal communication like whale and elephant communication. "The only reasonable way to account for …" is just claimed but not explained/justified. The statement just asserts a few things without actually arguing which requires explanation. It should be removed. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:54, 17 March 2025 (UTC) ::: People apparently do discuss the matter, e.g. in https://www.jstor.org/stable/10.1086/424978 The Argument from Language and the Existence of God by Jeffery L. Johnson and Joyclynn Potter, 2005. A physicalist may find it unconvincing but this debate is not only for physicalists. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 23:00, 17 March 2025 (UTC) ::: Other reading: https://answersingenesis.org/is-god-real/linguistic-argument-gods-existence/ A Linguistic Argument for God’s Existence by Dr. John Baumgardner and Dr. Jeremy D. Lyon on January 21, 2017. Some sentences seem to be taken from this source, and if so, they should be in quotation marks. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 23:05, 17 March 2025 (UTC) ::::Regarding both of your comments: that doesn't make it a coherent argument. The argument needs to be phrased as a coherent argument. A coherent argument looks for example roughly like so: "X and Y are so and so and because this implies Z, it means that V is likely since V can explain this as it can do W." An incoherent argument looks like this: "Language plays an integral role in the laws of nature and of DNA. As humans speak, God exists." It's not explained and needs to be removed for not being an argument. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 23:40, 17 March 2025 (UTC) ::::: Let me quote the current guideline: [[Wikidebate/Guidelines]]: "Unlike other debate systems, wikidebates are not aggregates of posts by different users, but a collaborative effort to compile and organize all arguments on a given issue." I emphasize "all arguments"; that is not to be read literally, but it does suggest that the bar for argument inclusion is low. Thus, even bad argument or argument that some find incoherent can be included; objections can then explain what is wrong with the argument. For instance, I included some arguments that I find utterly ridiculous in [[Are natural resources finite?]], and raised objections to them. The [[Are natural resources finite?]] wikidebate is a spectacle of absurdity (to my mind), but as long as the cornucopians such as Julian Simon and others are going strong, such debate does take place outside of wikidebates and is also well suited for wikidebates. ::::: What does [[User:Sophivorus|Sophivorus]], the creator/initiator of the wikidebate format, think about the matter? Should this particular argument that the existence of human language has to be explained by God be removed? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:26, 18 March 2025 (UTC) ::::::I have no issue with bad arguments. The problem comes when it's not an argument which is what I mean when saying it needs to be somewhat coherent. It could be readded if it's turned into an actual argument. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 11:04, 18 March 2025 (UTC) ::::::: My position is that it is an argument, as bad as it may seem. Furthermore, my position is that you (Prototyperspective) do not have consensus for removing the argument, so the argument (or set of statements most of which are quoted from an article, if you will) stays until you gain some support from others for removing the argument/set of statements. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:19, 18 March 2025 (UTC) == Edits by ~2026-51461-9 == I think these latest edits should be reverted {{ping|Dan Polansky|Sophivorus}}. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:33, 10 March 2026 (UTC) :Agree. [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discusión]] • [[Special:Contributions/Sophivorus|contribs.]]) 20:02, 11 March 2026 (UTC) == Should this proof be added to the Pro list??? == Nothing is nothing; Nothing is nonexistence; Nowhere and a no time has nothing existed; Something has always existed everywhere; Nothing is made of nothing; Everything is made of something that has always existed everywhere; Nothing is the cause of nothing; Something is the cause of everything; Everything has a cause; Something is self-causal; Everything is made of the self-causal that has always existed everywhere; Reality is consciousness; Language is isomorphic to consciousness; To describe consciousness requires language; Thinking/Describing transforms consciousness; Consciousness is self-descriptive and self-deterministic. [[Special:Contributions/&#126;2026-51461-9|&#126;2026-51461-9]] ([[User talk:&#126;2026-51461-9|talk]]) 21:57, 24 March 2026 (UTC) :No because the project is '''clearly''' closed, per the notice on the page. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:38, 24 March 2026 (UTC) :No also for other reasons or if reopened because this is merely claims without proper or sufficient explanation (or source either for that matter). It's relatively incoherent I think. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 22:44, 25 March 2026 (UTC) o29j8flrtrquqjrtnt9fqx25182fuza 0 222435 2807087 2716238 2026-04-30T07:44:35Z ~2026-26102-00 3069420 2807087 wikitext text/x-wiki ==Inspiration== [[File:PantalaFlavescensTalakaveri.jpg|thumb|right|An aggregation of dragonflies, during static swarm (migration)]] The main inspiration of the Dragonfly Algorithm (DA) algorithm proposed in ''2015'' <ref> S. Mirjalili, "Dragonfly Algorithm: A New Meta-heuristic Optimization Technique for Solving Single-objective, Discrete, and Multi-objective Problems", Neural Computing and Applications, in press, 2015, DOI: http://dx.doi.org/10.1007/s00521-015-1920-1 </ref> originates from '''static and dyn'''amic swarming behaviours. These two swarming behaviours are very similar to the two main phases of optimization using meta-heuristics: exploration and exploitation. Dragonflies create sub swarms and fly over different areas in a static swarm, which is the main objective of the exploration phase. In the dynamic swarm, however, dragonflies fly in bigger swarms and along one direction, which is favourable in the exploitation phase. [[File:DA Reynold.png|thumb|left|upright|Five primitive principles of swarming]] For simulating the swarming behaviour of dragonflies, three primitive principles of swarming in insects proposed by Reynold <ref> Reynolds, Craig W. "Flocks, herds and schools: A distributed behavioral model." ACM SIGGRAPH computer graphics. Vol. 21. No. 4. ACM, 1987. </ref> as well as two other new concepts have been utilized: separation, alignment, cohesion, attraction to food source, distraction from enemies. These five concepts allow us to simulate the behaviour of dragonflies in both dynamic and static swarms. The DA algorithm is developed based on the framework of the Particle Swarm Optimization (PSO) algorithm, so there are two main vectors: step vector and position vector. These vectors store the movement directions/speed and position of dragonflies, respectively. The main equations for these two vectors are as follows: <math> \Delta X_{t+1} = (sS_i + aA_i + cC_i + fF_I + eE_I) + w\Delta X_{t} </math> where <math>s</math> shows the separation weight, <math>S_i</math> indicates the separation of the i-th individual, <math>a</math> is the alignment weight, <math>A</math> is the alignment of i-th individual, <math>c</math> indicates the cohesion weight, <math>C_i</math> is the cohesion of the i-th individual, <math>f</math> is the food factor, <math>F_i</math> is the food source of the i-th individual, <math>e</math> is the enemy factor, <math>E_i</math> is the position of enemy of the i-th individual, <math>w</math> is the inertia weight, and <math>t</math> is the iteration counter. The equations for S, A, C, F, and E are as follows: [[File:DA Fig3.gif|thumb|right|Swarm simulation of dragonflies when s=0.1, a=0.1, c=0.7, f=1, and e=1. Note that the green asterisk is food source, the red asterisk indicates the enemy, black circles are individuals, and blue lines are the step vector of the dragonflies]] <math> S_i = - \sum_{j=1}^{N} (X - X_j) </math> <math> A_i = \frac{\sum_{j=1} ^ {N} X_j }{N} </math> <math> C_i = \frac{\sum_{j=1} ^ {N} X_j }{N} - X </math> <math> F_i = X^{+} - X </math> <math> E_i = X^{-} + X </math> where <math>X</math> is the position of the current dragonfly, <math>X^+</math> shows the position of a food source, <math>X^-</math> is the position of an enemy, <math>N</math> is number of neibouring dragonfly, and <math>X_j</math> indicates the position of the j-th neighbouring solution With the step vector, the position of dragonflies are updated with the following equation: <math> X_{t+1} = X_{t} + \Delta X_{t+1} </math> The right figure shows how the proposed model moves the individuals around the search space with respect to each other as well as food source and enemy. Please note that the green asterisk is food source, the red asterisk indicates the enemy, black circles are individuals, and blue lines are the step vector of the dragonflies. To see the animation more than once, you need to click on the figure. With the parameters(s, a, c, f, and e), we are able to simulate different swarming behaviour. For the figure, s=0.1, a=0.1, c=0.7, f=1, and e=1 have been utilized. The flying speed of dragonflies slowed down because w was linearly decreased from 0.9 to 0.4. Otherwise, dragonflies only explore the search space without convergence towards a point (exploitation). Note that there is no random walk in the movement of alone individuals in the above figure, while in DA, an individual moves using a random walk if there is no neighbouring solution at all. ==DA algorithm== The DA algorithm has been proposed for solving single-objective optimization problems. The Pseudo codes of this algorithm are as follows (for the equations' details, please refer to the paper): '''Initialize''' the dragonflies population Xi (i = 1, 2, ..., n) '''Initialize''' step vectors ΔXi (i = 1, 2, ..., n) '''while''' the end condition is not satisfied '''Calculate''' the objective values of all dragonflies '''Update''' the food source and enemy '''Update''' w, s, a, c, f, and e '''Calculate''' S, A, C, F, and E using Eqs. (3.1) to (3.5) in the paper (or above the page) '''Update''' neighbouring radius '''if''' a dragonfly has at least one neighbouring dragonfly '''Update''' velocity vector using Eq. (3.6) in the paper (or above the page) '''Update''' position vector using Eq. (3.7) in the paper (or above the page) '''else''' '''Update''' position vector using Eq. (3.8) in the paper (or above the page) '''end if''' '''Check and correct''' the new positions based on the boundaries of variables '''end while''' ==Binary Dragonfly Algorithm (BDA)== [[File:DA transfer function.png|thumb|right|a v-shapedTransfer function]] Since the DA algorithm is only able to solve continuous problems, it should be changed to solve binary problems. Basically, in discrete binary spaces, the position updating means switching between 0 and 1 values. The binary version of this algorithm has been name Binary DA (BDA), which is suitable for solving discrete problems. In order to do this, a v-shaped transfer function has been used. A transfer function maps a continuous search space to a binary. Transfer functions are computationally cheap tools for converting a continuous algorithm to a binary one. Such functions define the probability of changing the elements of a position vector from 0 to 1 or vice versa. The transfer function that has been used is illustrated on the right (for the equation, please refer to the paper). After all the pseudo codes of the BDA algorithm are as follows: '''Initialize''' the dragonflies population Xi (i = 1, 2, ..., n) '''Initialize''' step vectors ΔXi (i = 1, 2, ..., n) '''while''' the end condition is not satisfied '''Calculate''' the objective values of all dragonflies '''Update''' the food source and enemy '''Update''' w, s, a, c, f, and e '''Calculate''' S, A, C, F, and E using Eqs. (3.1) to (3.5) in the paper (or above the page) '''Update''' step vectors using Eq. (3.6) in the paper (or above the page) '''Calculate''' the probabilities using Eq. (3.11) in the paper '''Update''' position vectors using Eq. (3.12) in the paper (or above the page) '''end while''' == Multi-objective Dragonfly Algorithm (MDA) == Solving a multi-objective problem using a meta-heuristic require special considerations. In contrary to single-objective optimization, there is no single solution when considering multiple objectives as the goal of the optimization process. In this case, a set of solutions, which represents various trade-offs between the objectives, includes optimal solutions of a multi-objective problem. Before 1984, mathematical multi-objective optimization techniques were popular among researchers in different fields of study such as applied mathematics, operation research, and computer science. Since the majority of the conventional approaches (including deterministic methods) suffered from stagnation in local optima, however, such techniques were not applicable as there are not nowadays. This is the reason why stochastic optimization algorithms are reliable alternative due the high local optima avoidance. As mentioned above there is no more single solution for a multi-objective problem. The existence of multiple objectives prevents us from comparing the solutions with relational operators such as >, <, etc. Therefore, we have to use the definition of Pareto optimality to compare solutions. In this case, a solution is better than (dominates) another solution if and only if it shows better or equal objective value on all of the objectives and provides a better value in at least one of the objective functions. The answer for such problems is a set of solutions called Pareto optimal solutions set. This set includes Pareto optimal solutions that represents the best trade-offs between the objectives. In order to solve multi-objective problems using meta-heuristics, an archive (repository) is widely used in the literature to maintain the Pareto optimal solutions during optimization. Two key points in finding a proper set of Pareto optimal solutions for a given problem are convergence and coverage. Convergence refers to the ability of a multi-objective algorithm in determining accurate approximations of Pareto optimal solutions. Coverage is the distribution of the obtained Pareto optimal solutions along the objectives. Since most of the current multi-objective algorithms in the literature are of ''a posteriori'' type, the coverage and number of solutions are very important for decision making after the optimization process. The ultimate goal for a multi-objective optimizer is to find the most accurate approximation of true Pareto optimal solutions (convergence) with uniform distributions (coverage) across all objectives. For solving multi-objective problems using the DA algorithm, it is first equipped with an archive to store and retrieve the best approximations of the true Pareto optimal solutions during optimization. The updating position of search agents is identical to that of DA, but the food sources are selected from the archive. In order to find a well-spread Pareto optimal front, a food source is chosen from the least populated region of the obtained Pareto optimal front, similarly to the Multi-Objective Particle Swarm Optimization (MOPSO) <ref>Coello, Carlos A. Coello, and Maximino Salazar Lechuga. "MOPSO: A proposal for multiple objective particle swarm optimization." Evolutionary Computation, 2002. CEC'02. Proceedings of the 2002 Congress on. Vol. 2. IEEE, 2002.</ref> algorithm in the literature. To find the least populated area of the Pareto optimal front, the search space should be segmented. This is done by finding the best and worst objectives of Pareto optimal solutions obtained, defining a hyper sphere to cover all the solutions, and dividing the hyper spheres to equal sub hyper spheres in each iteration. After the creation of segments, the selection is done by a roulette-wheel mechanism. This mechanism allows the MODA algorithm to have higher probability of choosing food sources from the less populated segments. Therefore, the artificial dragonflies will be encouraged to fly around such regions and improve the distribution of the whole Pareto optimal front. For selecting enemies from the archive, however, the worst (most populated) hyper sphere should be chosen in order to discourage the artificial dragonflies from searching around non-promising crowded areas. The selection is done by a roulette-wheel mechanism again. The conceptual model of the best hyper spheres for selecting a food source or removing a solution from the archive are shown in the right figure. [[File:DA SelectionMechanism.png|thumb|right|Multi-objective optimization]] The archive should be updated regularly in each iteration and may become full during optimization. Therefore, there should be a mechanism to manage the archive. If a solution is dominated by at least one of the archive residences, it should be prevented from entering the archive. If a solution dominates some of the Pareto optimal solutions in the archive, they all should be removed from the archive, and the solution should be allowed to enter the archive. If a solution is non-dominated with respect to all of the solutions in the archive, it should be added to the archive. If the archive is full, one or more than one solutions may be removed from the most populated segments to accommodate new solution(s) in the archive. These rules are taken from the original MOPSO paper written by Professor Coello Coello and his colleagues <ref>Coello, Carlos A. Coello, Gregorio Toscano Pulido, and M. Salazar Lechuga. "Handling multiple objectives with particle swarm optimization." Evolutionary Computation, IEEE Transactions on 8.3 (2004): 256-279.</ref>. The above figure shows the best candidate hyper sphere (segments) to remove solutions (enemies) from in case the archive become full. All the parameters of the MODA algorithm are identical to those of the DA algorithm except two new parameters for defining the maximum number of hyper spheres and archive size. After all, the pseudo codes of MODA are as follows (for the equations, please refer to the paper): '''Initialize''' the dragonflies population Xi (i = 1, 2, ..., n) '''Initialize''' step vectors ΔXi (i = 1, 2, ..., n) '''Define''' the maximum number of hyper spheres (segments) '''Define''' the archive size '''while''' the end condition is not satisfied '''Calculate''' the objective values of all dragonflies '''Find''' the non-dominated solutions '''Update''' the archive with respect to the obtained non-dominated solutions '''if''' the archive is full '''Run''' the archive maintenance mechanism to omit one of the current archive members '''Add''' the new solution to the archive '''end if''' '''if''' any of the new added solutions to the archive is located outside the hyper spheres Update and re-position all of the hyper spheres to cover the new solution(s) '''end if''' '''Select''' a food source from archive: <math>X^+ </math>=SelectFood(archive) '''Select''' an enemy from archive: <math>X^- </math>=SelectEnemy(archive) '''Update''' step vectors using Eq. (3.11) in the paper (or above the page) '''Update''' position vectors using Eq. (3.12) in the paper (or above the page) '''Check''' and correct the new positions based on the boundaries of variables '''end while''' == References == <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags, these references will then appear here automatically --> {{Reflist}} == External links == * [http://au.mathworks.com/matlabcentral/fileexchange/51035-da--dragonfly-algorithm Source codes of DA in Matlab or Octave] * [http://au.mathworks.com/matlabcentral/fileexchange/51032-bda--binary-dragonfly-algorithm Source codes of BDA in Matlab or Octave] * [http://au.mathworks.com/matlabcentral/fileexchange/51033-moda--multi-objective-dragonfly-algorithm Source codes of MODA in Matlab or Octave] * [http://au.mathworks.com/matlabcentral/fileexchange/51031-dragonfly-algorithm-toolbox A toolbox for DA in Matlab or Octave] <!--- Categories ---> [[Category:Articles created via the Article Wizard]] c52z578r0pwn82n1j6g8cn4ro3vw8h7 102 270595 2807001 2806964 2026-04-29T15:22:08Z ~2026-26210-02 3069297 Candidature of Flavia Mori Sarti to the CS-DC Executive Committee in 2026 2807001 wikitext text/x-wiki <nowiki>**</nowiki>Please ''login'' in Wikiversity and then use the ''<nowiki/>'edit''' button: your edition mode will be 'WYSIWYG'. Each Candidature with its manifesto can take inspiration from those for previous years. ''Decision of the 2025 General Assembly 27th April:'' ''The 2026 elections are starting the 27th April, the date of the General Assembly:'' ''(i) There will be an election for Vice-President starting the 27 April 2026.'' ''(ii) To save time and effort - Instead of an election for 1/3 of the Executive Committee'' ''(EC) for 2025 we propose to keep the existing EC and invite members of the Council '''not on the EC''' to offer themselves as members of an enlarged Executive Committee. If less than six people propose themselves, they will be co-opted onto the Executive Committee. If more than six propose themselves we will hold an election for 1/3 EC.'' The deadline for your candidature is Saturday 16th May 2026 at 24:00 CET. == Candidature for the New Vice-President for [2026, 2027] == '''Manifesto – Candidature for New Vice-President of UNESCO UniTwin CS-DC''' == New Candidatures to the Executive Committee for [2026] == === Manifesto – Candidature for the Executive Committee of UNESCO UniTwin CS-DC === '''Flavia Mori SARTI, Ph.D., Professor and Researcher, University of Sao Paulo, Brazil''' I present my interest in being candidate for the CS-DC Executive Committee to contribute to the dissemination of knowledge in complex systems at the local and global level. I am representative of the University of Sao Paulo (USP) at the CS-DC since 2012. I work with complex systems modeling since the creation of the Interdisciplinary Research Group of Complex Systems Modelling at the School of Arts, Sciences and Humanities (EACH-USP) in 2006. Our research group succesfully implemented the first interdisciplinary graduate program titled Master in Complex Systems Modeling in Brazil, in 2010. I have been participating in the program commission since 2010, and I was coordinator of the graduate program from 2010 to 2014. I have supervised 12 students in the program and two postdoctoral fellows, followed by publication of several book chapters and papers on complex systems applied to public policies, including models on tax evasion, health systems regulation, health insurance, food policy and nutrition programs, and complex networks on scientific collaboration and global food trade. I have been invited to present seminars and talks on complex systems applied to food systems, health economics, and public policy. My goals in the CS-DC Executive Committee include: * To disseminate the role of CS-DC in education and research on complex systems, especially in developing countries; * To support and to engage other research groups working with complex systems for participation in the CS-DC; * To contribute further with management and organization of CS-DC activities during the period of 2022-2024; * To continue supporting capacity building in complex systems through supervision and other academic activities; * To foster innovation, research and development strategies through complex systems applications in public policy and entrepreneurship. == Candidature for the New President for [2025, 2026] == '''Manifesto – Candidature for New President of UNESCO UniTwin CS-DC''' '''Prof. Paul Bourgine''' : If elected, my main commitment is to create the conditions for a self-organized development of our UNESCO UniTwin CS-DC as autonomous communities of communities for our flagship TIMES and its Knowledge & Knowhow Accelerator one-for-all & all-for-one (KKA). We know now how to realize —for the two above commitments— the 3^rd UNESCO commitment, i.e., the ‘computational ecosystem’. It will use the mature part of Web 3.0, especially the InterPlanetary File System (IPFS). Thanks to our previous efforts especially of the two last years, the remaining work amount is ten times less than we were anticipating at the beginning of the 2^nd renewal of our UniTwin by UNESCO 2020-2026.   If elected, my duty will be not only to fulfill entirely the commitments of our Cooperation Program with UNESCO but also starting an exponential increasing development wave for our UniTwin network (through their continent and country Councils) and of our e-Campus (through CS-DC’25 and e-Labs’26 Conferences especially for our Flagships for sustainable development). The Knowledge & Knowhow Accelerator will directly benefit from 1) such conference series, 2) our past and new flagships for sustainable development and 3) a new decentralized strategy for collecting donations in our decentralized network of X-Legal Entities. == New Candidatures to the Executive Committee for [2025] == === Manifesto – Candidature for the Executive Committee of UNESCO UniTwin CS-DC === '''Prof. Silvius STANCIU, PhD in Economics, PhD in Engineering, Habil.''' Full Professor, “Dunărea de Jos” University of Galați (UDJG), Romania Editor-in-Chief, ''Journal of Agriculture and Rural Development Studies (JARDS)'' Former Vice-Rector, Former Director of DFCTT and CTT UGAL ----'''Dear Councillors,''' It is a great honor for me to submit my candidacy for the Executive Committee of the UNESCO UniTwin Complex Systems Digital Campus (CS-DC). With more than '''30 years of experience in academia''', I am currently Full Professor and doctoral advisor at “Dunărea de Jos” University of Galați (UDJG), Romania — a public research university with a strong regional impact and a long-standing tradition in interdisciplinary education and innovation. I hold two doctoral degrees — one in Economics and one in Engineering — and I am a habilitated professor. I have published '''163 ISI-indexed scientific articles''' and have a '''Clarivate H-index of 14'''. My research focuses on '''food security, circular economy, technological innovation, rural development''', and '''complex systems in agro-food value chains'''. I am the founder and coordinator of Romania’s first doctoral program in ''Engineering and Management in Agriculture and Rural Development (IMADR)'', with '''9 PhD graduates''' and '''9 doctoral students''' currently under my supervision. I also serve as '''Editor-in-Chief''' of the ''Journal of Agriculture and Rural Development Studies (JARDS)'', dedicated to interdisciplinary research in sustainable food and rural systems. Over the past decade, I have been involved as director or expert in '''more than 45 national and international research projects''', including Horizon-compatible initiatives and cross-border cooperation programs. I coordinated a '''Romania–Republic of Moldova cross-border project''' (2020–2021) and currently lead '''two new ROMD-funded projects''' entering implementation. My former institutional leadership roles include: * '''Vice-Rector for Research and Innovation''' * '''Director of the Department for Institutional Development (DFCTT)''' and of the '''Technology Transfer Center (CTT UGAL)''' * '''Member of national and international quality and research bodies''', including CNATDCU, ARACIS, and CMPTJ ----'''If elected, I am committed to:''' * Expanding the CS-DC network in '''Eastern Europe and the Black Sea region''', enhancing scientific and territorial diversity; * Supporting '''POEM''' and '''FOOD flagship programs''' through digital education, doctoral/postdoctoral collaboration, and innovation ecosystems; * Promoting '''open science''', international e-seminars, and interdisciplinary MOOCs; * Coordinating thematic initiatives in '''agro-complexity, food systems resilience''', and '''sustainable rural innovation'''. As a representative of a '''UniTwin member institution''', I see this candidacy as a unique opportunity to strengthen UDJG’s role within the CS-DC ecosystem. I fully embrace the CS-DC mission to foster global collaboration, education, and research in complexity science. I am ready to bring '''vision, experience, and energy''' to the Executive Committee and help shape the future of our UniTwin community. ----'''Sincerely,''' '''Prof. Silvius STANCIU, PhD in Economics, PhD in Engineering, Habil.''' Representative of “Dunărea de Jos” University of Galați (UDJG) '''Professional Identifiers:''' * Web of Science Author ID: R-8246-2017 * ORCID: 0000-0001-7697-0968 * Scopus ID: 36633317700 * Google Scholar: Silvius Stanciu * ResearchGate: Silvius Stanciu == New Candidatures to the Executive Committee for [2024] == '''Enver Oruro Puma, Ph.D., Principal Investigator of Neurocomputing, Social Simulation, and Complex Systems Laboratory at the Instituto Científico of Universidad Andina del Cusco, Peru''' Dear Councillor of the UNESCO UniTwin CS-DC. I am very honored to place my candidature for the UNESCO UniTwin CS-DC Executive Committee. I am Enver Miguel Oruro Puma, Ph.D., principal investigator of Neurocomputing, Social Simulation, and Complex Systems Laboratory at the Instituto Científico of Universidad Andina del Cusco, Peru (https://sites.google.com/view/orurolab/). Since 2009, I have promoted and organized conferences and academic events on Complex Systems in Latin America. Recently, I have promoted the area of computational neuroscience on infant attachment (https://sites.google.com/view/envermiguel/seminar-in-maternal-infant-relationship-studies). It would be a great honor for me if given the opportunity to contribute to the Executive Committee of UNESCO UniTwin CS-DC in the integration of Complex System research groups in the Latin American Region. For this, I propose the creation of two periodical activities: 1) A Special Lectures Series on Complex Systems UNESCO UniTwin CS-DC oriented to experts on Complex Systems, and 2) A Invited Advanced Lectures UNESCO UniTwin CS-DC oriented to experts who do not identify explicitly with complex systems '''Pierre Collet, full professor of Strasbourg University, on secondment to Universidad Andrés Bello, Instituto de Tecnología para la Innovación en Salud y Bienestar, Viña del Mar, Valparaiso, Chile'''. Since 2012, I have contributed to the elaboration of the CS-DC Unesco UniTwin together with Paul Bourigne, Jeffrey Johnson and many others, and I have been co-coordinator of the CS-DC UniTwin with Cyrille Bertelle since its creation in 2014. Starting part of this great adventure has changed my academic and personal life: thanks to the UniTwin, I have changed my research from stochastic optimisation, artificial evolution and AI in general to complex systems and epistemology. Participating in this UniTwin allowed me to make new contacts and start incredible projects that I could not have imagined before. It has even changed my life, as I am now living in Chile, having been recruited by ITISB, an institute founded by Carla Taramasco, the CS-DC representative for South America. Together with Paul and others, we would like to revive UniTwin by preparing another world conference inspired by the great success of [https://cs-dc-15.org CS-DC'15] and also develop flagship projects such as POEM (Personalised Open Education for the Masses) and the [https://en.wikiversity.org/wiki/Portal:Complex_Systems_Digital_Campus/E-Laboratory_on_complex_computational_ecosystems ECCE e-lab], which this year has welcomed a new very active [[Figures of Play/Les figures du Jeu e-team|Figures of Play]] that has started the [https://ludocorpus.org/ Ludocorpus] in France. As said before, this incredible UniTwin adventure always pays off for those who invest in it and in its great challenge: to develop the new science of complex systems through research and education. Through its projects, it contributes to making the world a better place to live in, despite the constant attacks on science coming from the most unlikely places. Science is the solution, not the problem, to many of the world's plagues. We must put our energy into developing it and defend it against all its detractors. That is why I am once again standing for election to the Executive Committee of this great CS-DC UniTwin. Modern science is Complex Systems science. It is important that its beacon continues to illuminate the world, and we must invest our time and energy in it. == Candidature Deputy President for [2024, 2025] == '''Jeffrey JOHNSON, Professor of Complexity Science and Design, The Open University, UK''' I offer myself as a candidate both to be President and to the Executive Committee of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC) so that I can help to drive it forward to achieve it goals. I am particularly committed to our educational efforts. I have made four MOOCs on the FutureLearn Platform for CS-DC ( <nowiki>https://www.futurelearn.com/partners/unesco-unitwin-complex-systems-digital-campus</nowiki> ): Global Systems Science (2015-16); Systems Thinking and Complexity (2017-18); First Steps in Data Science with Google Analytics (2018-19) and COVID-19 - Pandemics, Modelling and Policy (2020). CS-DC has a great opportunity to become the global university providing interdisciplinary education for a better world. I am also committed to our research mission with UNESCO towards the achieving the U.N. Sustainable Development Goals. My own research on representing the dynamics of complex multilevel systems is relevant to many of the research initiatives of CS-DC. I have extensive experience working within the complex systems community. I have run various coordination actions supporting research programmes funded by the European Commission, I am a founder member and past president of the Complex Systems Society, and I am Deputy-President of the CS-DC. I believe this experience will enable me to make a significant contribution the CS-DC over the next three years. == New Elected President for [2023, 2024] == Paul BOURGINE, present President of the UNESCO UniTwin CS-DC, Complex Systems Institute of Paris I offer myself as a candidate to be President of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC). My previous commitment two years ago is below. The bad news is that it was not achieved. The good new is that we know now how to create 'autonomous community of autonomous communities' as a social network with IPFS (the InterPlanetary File System) like the new development of Wikipedia. If elected, my first commitment is to finish this job as quickly as possible. My second commitment is simultaneously to visit each country of the UniTwin for creating its country.CS-DC and its roadmap with young eTeams shared by their Universities with a senior scientific committee. The eTeam projects will have the opportunity to be submitted to the EU calls or other ones. Enver Oruro PhD, Head of Neurocomputing, Social Simulation and Complex Systems Laboratory, Universidad Andina del Cusco, Peru. '''I would like to nominate Professor Paul Bourgine.''' == New Elected Members to the Executive Committee for [2022,2023, 2024,2025] == '''2Dr Mohamed Abdellahi (Ould BABAH) Ebbe, Mauritania,''' * Senior Advisor for the CILSS Executif Secretary for international Partnership and formal General Director of the Institut du Sahel/CILSS www.insah.org; ·       Commissionaire General of CILSS for Horticulture Universal  Expo of DOHA 2023-2024   <nowiki>https://www.dohaexpo2023.gov.qa/en/</nowiki> with central thems: '''CENTRAL THEME: GREEN DESERT, BETTER ENVIRONMENT''' * Executive Director of the Orthopterist Society (400 researchers among the globe) <nowiki>https://orthsoc.org/</nowiki> * We have organized our last congress during 16-20 0ctober in Merida Mexique <nowiki>https://ico2023mexico.com/</nowiki> By obtaining the honor of having your hoped-for confidence for continuing this post of member of the executive council of the CS-DC, I will work, in priority and in the short term on two main subjects: ## '''The transboundary plague of the Desert Locust (Schistocerca gregaria (Forska l , 1775))''' This plague of the Desert Locust of more than 3000 years that cites all our holy books (the Tourah, the Bible and the Koran) and which continues to be present to this day and to wreak devastating devastation. In case of invasion, it can affect the agriculture and pastures of about 25 countries including those of the poorest countries of the world, from Mauritania to India, while its best and most effective strategy of struggle is preventive struggle by targeting its first centers of gregarization which are very small in space and much better known today. In 2005, the costs of its struggle in the Sahel and North Africa amounted to half a billion dollars, with 8 million farmers and pastoralists affected in the Sahel. It also massively invaded Asia and Africa. 'East Africa in 2020. On this subject, I have spent 30 years studying and fighting and developing a national strategy against this scourge which has made it possible to establish a whole prevention model and an institutional, technical, operational mechanism. and scientific effective in my country that can be adapted and copied and in all other affected countries: Biogeography of the desert locust Schistocerca gregaria, Forskal, 1775: Identification, characterization and originality of a gregarious focus in central Mauritania (HR.HORS COLLEC.) (French Edition) - Babah Ebbe, Mohamed Abdallahi | 9782705670573 | Amazon.com.au | Books <nowiki>http://www.worldbank.org/en/news/feature/2010/01/07/improved-ways-to-prevent-the-desert-locust-in-mauritania-and-the-sahel</nowiki>, http: // whatsnext.blogs.cnn.com/2012/02/02/in-mauritania-sunny-with-a-chance-of-locusts/ I was invited last year by Royal Society 20-21 may 2024 to moderate one session on locust research management (La plasticité des criquets et des abeilles dans un monde en mutation | Société royale) and in a “International Conference on New Technology and Concepts for Sustainable Management of Locusts and Grasshoppers” held from 2 to 7 June 2024 in Jinan, Shandong, China.We are also preparing our Orthopterist congress in Argentina during the next mars 2026 <nowiki>https://ico2026.com.ar</nowiki> '''All this is in addition of more than 110 publications or joint publications on the locust, its environment and management''' # '''Senior Adviser to the CILSS Executive Secretary for International Partnerships'''] [Assistance to Mauritania (or 3 months) in the preparation of the organisation of the Nouakchott+10 High-Level Forum on pastoralism held in Nouakchott from 6 to 8 November 2024, various advising for the international partnership and the mobilization of resources including preparation of the organization of a round table planned in OPEC Vienna Austria for the mobilization of Arabic and Islamic funds for the financing of the CILSS 2050 strategic plan # '''The Sahel Institute (INSAH) www.insah.org of the Permanent Interstate Committee for Drought Control (CILSS)''' that I lead and which has been doing extraordinary work for almost half a century in the field of research and development of animal and plant production techniques and also in the field of support for demographic, population and development policies, in favor of the populations of our 13 Sahelian, coastal and island member countries. This work covered the majority of good practice technologies in the field of plant and animal production, natural resource management, land rstauration, cultivation techniques, post-harvest, machining, dehulling operations technology. / ginning, Conservation and storage, good resilience practices Research on the demographic dividend, gender and the empowerment of women and the Population / Development interrelations ... etc The results of all this work are contained in a database. data, online <nowiki>http://publications.insah.org/</nowiki>, containing more than 1,500 books, scientific and technical articles that will have to be modernized and connected to the CS Meta data. As General Commissionaire of CILSS for Horticulture Universal  Expo of DOHA 2023-2024   <nowiki>https://www.dohaexpo2023.gov.qa/en/</nowiki> with central thems: '''CENTRAL THEME: GREEN DESERT, BETTER ENVIRONMENT''' I am working  in introducing as detailed below: '''CILSS ''contribution to the improvement of sustainable horticultural agricultural production in a context of drought''''' '''I. PRESENTATION OF THE EXPO''' Expo 2023 in Doha is part of the fight against desertification. The Expo will be held from 2 October 2023 to 28 March 2024 under the theme "'''''Green Desert, Better Environment'''''". The aim is to encourage, inspire and inform people about innovative solutions to reduce desertification. The exhibition will provide an international platform for participants, stakeholders, decision-makers, nongovernmental organizations and experts to address the global challenge of "desertification", while making a valuable contribution to achieving a sustainable future. During the 6 months of the Expo, nearly 3 million visitors from over 80 countries are expected The objectives of this Expo are in line with those of the CILSS, which seeks to improve the living conditions of the people of the Sahel in a sustainable manner. This is why the participation of CILSS in this Expo is important for the region and its vulnerable populations. '''OBJECTIVES OF EXPO 2023 DOHA, QATAR''' Expo 2023 Doha, Qatar is defined by the following objectives: -   Encourage horticultural innovation by focusing on Qatar's climate, water and soil. -   Promote Expo 2023 in Doha, Qatar, as a catalyst for international investment and business opportunities. -   To propose innovative actions that would allow humanity to fight against desertification more quickly and decisively before it is too late. -   To build up useful environmental outputs for future generations. '''II. ORGANISATION OF THE CILSS PARTICIPATION''' '''II.1. GOALS OF CILSS EXHIBITION:''' 1. Sharing experiences and best practices, 2. Building International Partnership, 3. Promoting technology and innovation Finally, I will continue to work actively with my colleagues on the Executive Board on all aspects of other cross-border scourges but also all aspects of improving agro-sylvo-pastoral production '''Dr. Xabier E. Barandiaran, Lecturer at the University of the Basque Country (UPV/EHU), Department of Philosophy, Donostia - San Sebastian, Spain''' I would like to present [https://xabier.barandiaran.net myself] as a candidate for the Executive Committee. I have been the representative and coordinator between CS-DC and the [https://ehu.eus University of the Basque Country] since 2013. I develop my academic research at the [https://ias-research.net IAS-Research Centre for Life, Mind, and Society], with a focus on the understanding of autonomous and complex adaptive systems (from biology to cognition, from brains to societies). I am the author of over 50 indexed publications on topics related to complex systems, philosophy of mind, complex epistemology, simulation models of the origins of life, minimal agency, evolutionary robotics, complex social network analysis, etc. I recently received the “Award for Distinguished Early-Career Investigator” by the International Society for Artificial Life. Overal I have been awarded with 7 different grants and have actively participated on 15 different research projects. I have also supervised 2 PhD thesis (4 more still in development) and I hold an extensive record of scientific and innovative management experience in different academic and public institutions as founder of research networks ReteCog.Net and FLOK Society – Buen Conocer and head of RDI at Barcelona City Council (2016-2018). I have also organized several national workshops, summer schools and conferences, and 2 international summer schools, 4 international workshops and one international conference. I am currently the Principal Investigator of a founded research project (with more than 30 research-collaborators) on a complex systems' approach to the concept of autonomy beyond its classical conception as an individual bounded property. As part of my university's goal of fostering international collaboration and opening up e-learning and research initiatives I would like to get more deeply involved on CS-DC with the following goals: * To desing the infrastructure, learning-experience, research-experience and content for distributed, open access and high-quality digital campus facilities. * To involve local agents (student, teachers, researchers and institutions) on the initiative of the network. * To foster collaboration, co-production and resource sharing between teaching and research facilities between priviledged richer countries and lower-income ones. In particular, but not exclusively, and for obvious reasons related to sharing the same language, to foster ''collaboration between European and Latin-american universities'', research initiatives and students through CS-DC. * To develop at least one ''prototype'' of a MSc level online course (and research network module) around complex cognitive systems that can serve as a model for the other fields of the network. * To develop a clear conceptual and communicative framework for CS-DC to be able to attract more participants, resources and broader attention and success as pioneering international initiative. '''Dr. habil. László Barna Iantovics, Professor at “George Emil Palade” Univ. of Medicine, Pharmacy, Science and Technology of Tg. Mures, Romania''' With the present manifesto, I would like to be a candidate for the CS-DC Executive Committee. I have been the representative of “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures from Romania in CS-DC by many years. Some of my research and academic activities were related to the complex systems, including: publications; organized conferences (e.g. Symposium on Understanding Intelligent and Complex Systems - UICS 2009; 1st Int. Conf. on Complexity and Intelligence of the Artificial and Natural Complex Systems Medical Applications of the Complex Systems. Biomedical Computing -CANS 2008; 1st Int. Conf. on Bio-Inspired Computational Methods Used for Difficult Problems Solving. Development of Intelligent and Complex Systems - BICS 2008); membership in conference committees (e.g. Int. Conf. Emergent Properties in Natural and Artificial Complex Systems - EPNACS 2007; Workshop on Complex Systems and Self-organization Modeling -CoSSoM 2009); Journal Special Issues (e.g. Special Issue on Complexity in Sciences and Artificial Intelligence; Special Issue on Understanding Complex Systems); membership in Journal’s Editorial Boards (e.g. Complex Adaptive Systems Modeling -CASM, SpringerOpen), and contribution to research performed in projects and projects coordination (Social network of machines- SOON; Hybrid Medical Complex Systems -ComplexMediSys). I am the director of the Research Center on Artificial Intelligence, Data Science and Smart Engineering (Artemis). I would like to involve myself much deeper in the life and activities of the CS-DC community. My principal objectives are: * To involve junior and senior researchers from my university in activities regarding research and education related to complex systems. * To involve universities and research institutes to actively contribute to the CS-DC development. * To involve myself in the joint coordination with other CS-DC members of a doctoral and postdoctoral students’ group that will be involved in the CS-DC community works. * To strengthen the research direction with the theme: applications of intelligent complex systems and machine intelligence measuring. One of the subtopics of interest will be the application of complex systems, artificial intelligence and data science in medicine, pharmacology, and healthcare. '''Flavia Mori SARTI, Ph.D., Professor and Researcher, University of Sao Paulo, Brazil''' I would like to present my candidature for the CS-DC Executive Committee in the period 2022-2024 to contribute to the dissemination of Complex Systems Science. I have been representative of the University of Sao Paulo (USP) at the CS-DC since 2012, and I have been working with complex systems since the creation of the Interdisciplinary Research Group of Complex Systems Modelling at the School of Arts, Sciences and Humanities (EACH-USP) in 2006. Our research group succesfully implemented the first interdisciplinary graduate program (Master) in Complex Systems Modelling in Brazil, in 2010. I have been participating in the coordinating commission of the program since 2010, and I was coordinator of the graduate program from 2010 to 2014. I have supervised seven students in the Master program, which resulted in thesis, book chapters, and papers published on the subject of complex systems, including models on tax evasion, health systems regulation, food policy and nutrition programs, and complex networks on scientific collaboration and international food trade. I also contributed to the organization of the e-Session "Economics as a Complex Evolutionist System" on the CS-DC'15 World e-conference in 2015, and have been invited to present seminars on complex systems applied to health economics, health technology assessment, and public policy of nutrition and health. My goals in the CS-DC Executive Committee include: * To disseminate the role of CS-DC in education and research on Complex Systems, especially in Brazil and other developing countries; * To support and to engage other research groups working with Complex Systems for participation in the CS-DC; * To contribute further with management and organization of CS-DC activities during the period of 2022-2024; * To continue supporting capacity building in Complex Systems through the Complex Systems Modelling Program at USP; * To participate in innovation, research and development activities based on the application of Complex Systems in public policy and entrepreneurship. '''Pr Panos Argyarakis, Professor in the University of Thessaloniki, Greece.''' I have been with the Complex Systems Society since its inception in 2004 by participating in the NEST projects Dysonet and Giacs which created CSS. My experience in the Executive Committee will be to contribute towards the spreading of the Complexity idea to various levels of education throughout the different countries. I am currently the PI in an Erasmus+ network that introduces new models of teaching and investigating how is education been affected for future generations. I can contribute in decision making for such important activities, and also serve as liaison with the European Commission, and the Complex Systems Society, due to my past experience. I have extended organizational experience by organizing several internationally meetings in this field that were attended by large audiences. My research interests are related to Complex systems and Networks. Scale-free, random, and small world networks. Dynamic properties on networks, Diffusion, spreading phenomena on networks, disease spreading. Phase transitions, percolation model, reaction-diffusion processes, trapping processes. Random walks. '''Ali Moussaoui, Professor, University of Tlemcen, Department of Mathematics, Algeria,''' I wish to present my candidature to become member of the executive committee of the CS-DC, I wish to develop collaborations with the partner universities in the field of complex systems. I wish to participate in the creation of international mixed laboratories and international masters on complex systems. In the past, I was responsible for a master's degree entitled: modeling of complex systems in our department, I am currently responsible for a research team entitled: Modeling of complex systems in our laboratory, I was responsible for a Franco-Algerian project on the modeling of complex systems. My research skills are focused on the modeling of complex natural and biological systems. '''Carlos Gershenson, Research Professor, Universidad Nacional Autónoma de México.''' I was involved with CS-DC in its initial years in UNESCO's UniTwin, also representing UNAM. I have been editor-in-chief of Complexity Digest since 2007. I co-organized the Conference on Complex Systems in 2017. I am currently vice-president secretary of the Complex Systems Society (CSS). I am a strong proponent of open online learning. I managed to start a collaboration between UNAM and Coursera, which has led to more than a hundred MOOCs and millions of enrolled students. I would be interested in strengthening the relationship between CS-DC and CSS, as well as other organizations. ==Elected members to the Executive Committee for 2021 == '''Carlos J. BARRIOS H., PhD., Professor, Bucaramanga, Colombia ''' I write to express my interest to candidacy to be part of the CS-DC Executive Committee. I'm very motivated to develop actions to strengthen digital ecosystem supporting research and education proposals of our CS-DC Council. Among these years participating in the CS-DC group, I can see different ways to leverage the impact and the development of our actions with computational strategies, and now, I want to be part of the leadership council joined mutual visions. My experience leading the Advanced Computing System for Latin America and Caribbean (SCALAC : http://scalac.redclara.net ) and as member of other leadership boards in international projects (mainly between Europe and Latin America) supports my candidature. (linkedin.com/in/carlosjaimebh) Also, my role as professor, director and researcher contributes to build the common vision of the CS-DC Council and the leadership of the CS- DC Executive Committee. '''Mina TEICHER, Professor of Bar-Ilan University, Israël''' I submit my candidacy to the Executive Committee of the CS Digital Campus. If elected I will work towards our following needs, using my past experience in Professional international societies, universities managements and the data industry :  * We need in the near future to build an optimal and effective agreement with the Complex System Society. * We need to build a business  plan for fund raising. * We need to build a modular budget for 2021. * We need to build a strategy for geographically extension.  * We need to build a strategy for thematic  extension.  * We need to build partnerships with the big multi national high tech Companies in network and in content.  '''Yasmin MERALI, Professor of University of Hull, UK''' This manifesto is connected with the ideals that I had as a founding member of our UniNet which was conceived as part of the FP7 ASSYST project. CS-DC has come a long way since its initial conception. The way I see it, there are three categories that have grown to emerge as our core activities- * Capacity building through education and training in Complex Systems Science * The application of Complex Systems Science to address global challenges * The advancement of Complex Systems Science through research and development. I believe this is a good time to link back to the inception of our UNITWIN which was in part inspired by considerations of issues at a human scale, and the desire to address the inequalities that divided the so-called developed and developing countries. This resonates strongly with the ambition of the Sustainable Development Goals (SDGs) we are currently grappling with. In the growth phase of the UNITWIN and CSDC we have been focused on extending the size of the network, and scaling up our educational offerings across the digital campus. In the next phase I believe we need to: # understand and leverage the diversity and distinctive capabilities and resources (e.g. indigenous knowledge) of the countries in our network to develop a healthy ecosystem, and # tailor the support that we provide to align with the diverse nature of their relational and social capital and their economic, political and environmental challenges and priorities with regard to the SDGs. I am concerned that if we do not explicitly design a social/ideational exchange mechanism that attends to these two imperatives, we will not have full, active participation of all member institutions, and the countries of the South that do not currently have champions in Europe will be marginalized. If elected I would champion a strategy of organizing ourselves following the Complex Adaptive Systems paradigm, as a hyper network with dynamically connected local clusters. In practical terms I would like to begin by establishing the local (country-based) clusters and establishing a discourse that would allow us to map the diverse profiles, challenges and aspirations for the different countries. This would then form the basis for the development of a mechanism for shaping the meaningful collaborative development of our three core activities to deliver advances that are globally co-ordinated and locally responsive. Personal Profile: I am Professor of Systems Thinking at the University of Hull and have served as Director of the Centre Systems Studies there. Prior to that I was Co-Director of the Doctoral Training Centre for Complex Systems Science at the University of Warwick. My research is transdisciplinary, focusing on the use of Complex Systems Science to enhance the resilience of socio-economic systems. I am an Expert advisor to the EU and I have significant experience of lecturing internationally as Visiting Professor in Asia, Europe and the USA. '''Céline ROZENBLAT: Professor, University of Lausanne - Institut de géographie et de durabilité (IGD), Switzerland''' I'm pleased to applied to become member of the CS-DC council. As founding member of CS-DC, my university, the university of Lausanne, is very engaged in Complex sciences. I would not only represent my university, but also social science as geographer and vice-president of the International Geographical Union and member of the International Science Council commission on Urban Health and Well being. I would act in the council in specific programs to develop the reality of the Digital Campus of the Complex systems. All these actions combine very ambitious interdisciplinary approaches, and in this perspective, we developed with CS-DC for 3 years the TIMES Flagship Territorial Intelligence For Multilevel Equity And Sustainability. It comprises four main programs: '''SIRE''': Socially Intelligence Roadmap Ecosystem '''POLE:''' Personalized Open Lifelong Education '''WOSI:''' Worldwide Open Smart Innovation '''WOSP:''' Worldwide Open Stochastic Prediction In this perspective a MOOC « Healthy Urban system » is now in development, basing the interdisciplinary approach on the CS-DC Road-Map grid. It seems very useful and relevant in this implementation stage. I would help to develop other programs in this perspective\[Ellipsis] '''André TINDANO, Director General of CARFS (African Center for Research and Training in Synecoculture)''' What motivates me to aspire to the position of member of the executive committee of CS DC is my long term participation in the promotion of sustainable development and my commitment to the sharing of knowledge and expertise. My research interests Sustainable agriculture, ecology, nutrition, life science. I have a strong experience in: * Administration and management of development projects and programs; * Accompaniment of associations and groups; * Technical capacity building (animation of training sessions and reflection workshops). * Action research; * Sociological, socio-economic and economic studies. * Development of development projects and programs; * Training of trainers * Results Based Management Training (RBM) * Monitoring and evaluation of development programs and projects; * Management of programs and development projects; * Institutional development and organizational strengthening; * Development and implementation of training / awareness / animation program; * Very good knowledge of participatory methods '''Guiou KOBAYASHI, Associate Professor at Federal University of ABC in São Paulo State, Brazil. ''' I worked with fault-tolerant computer systems for nuclear power plants and Metro signaling systems and recently my interests have evolved to resilience properties of Complex Systems. Traditionally, redundancy was the main feature for fault-tolerant and fail-safe systems, but the adaptability and the evolution of Complex Systems are the key elements for the resilience of these systems. How to characterize, design and implement these key elements in our future resilient systems? The Complex System - Digital Campus (CS-DC) is a way to create a world-wide community of researchers, philosophers and students to promote and discuss this kind of questions involving Complex Systems. For me, participating in its foundation was a great honor and I am very glad for the opportunity that I have had to contribute since 2012 in the consolidation of CS-DC. Through this manifest I am applying to be one of the members of the new CS-DC's Executive Committee. I would like to help just a little more to strengthen and structure this fantastic community through which I had the opportunity to meet important people with very interesting works that expanded my knowledge of Complex Systems. Although my University and my personal contribution for CS-DC are very limited and small, I hope to continue to work with this great team. '''Pierre COLLET, Professor of Computer Science, University of Strasbourg, France. Co-coordinator of the CS-DC UNESCO UniTwin''' The CS-DC initiated by Paul Bourgine, Jeffrey Johnson, Cyrille Bertelle and many others has been an extraordinary adventure a) to instantiate as a UNESCO UniTwin and b) to develop and run since it was enacted in July 2014. Many a night have been spent on designing its inner workings, so that it can deliver an effective affordance for the scientists who wish to develop the science and teaching of Complex Systems. Indeed, many projects seeded in the CS-DC have come to fruition, showing the enormous potential of this fertile environment not only for research, but also for teaching: the BBB rooms set up by the CS-DC have not only made it possible for the CS-DC to organize conferences, but have also shown their potential as remote teaching rooms in many Universities around the world. It has been an honour for me to be part of the development of the CS-DC since its beginning, but so much remains to be done! In this manifesto, I hereby express my strong desire to continue developing the CS-DC in these trying times, when the effects of the pandemics stretch thin the social links that our research and teaching communities need most. My objectives for this new mandate are not only to deliver a new world conference (originally planned in 2020 but unfortunately delayed due to the high toll imposed on us all, teachers and researchers alike, by COVID-19) but also continue on developing not only efficient complex computational ecosystems (cheap powerful PARSEC machines have been installed in several universities) but more specifically remote teaching environments based on complex systems, to mitigate the terrible impact of the pandemic on face to face education, within the POEM CS-DC flagship, on which Paul Bourgine and myself have been working for many years now.. '''Mariana C. BROENS, Professor - UNESP - BRAZIL.''' As members of the Executive Committee, our main challenge will be to raise, analyse and to discuss possible positive/negative ethical and political implications of the further development of the Complex Systems Science, and their application on studies of everyday social problems. In particular, We believe that the widespread use of complex system models and Big Data analytics can bring important questions about people's privacy, personal and corporate responsibility, widespread surveillance by public or private institutions, among many others, that should be deeply discussed in our community. Our contribution will be to raise and deepen these discussions from an interdisciplinary perspective. '''Cyrille Bertelle, Professor in Computer Sciences, University Le Havre, France, co-coordinator of the CS-DC UNESCO UniTwin''' I am a candidate for the CS-DC Executive Committee to represent the University of Le Havre Normandy, which is co-coordinator with the University of Strasbourg of the convention of recognition of the CS-DC as UniTwin by UNESCO. The University of Le Havre has made available to the community, resources and skills to provide digital collaborative tools for the organization of the Councils and the CS-DC'15 virtual conference. The objective I wish to take is to facilitate the involvement of member universities not only by their representatives on the council but by allowing researchers from these member institutions to join concrete and accessible actions. Co-responsible in the past of a master's degree on complex systems and then of the creation of the institute on complex systems in Normandie (France) in a multidisciplinary framework, I have participated in the setting up of the project labeled by the French national program of investments for the future and entitled "Smart Port City". The aim is to think about the future of territories in a sustainable development approach supported by new technologies and concerned about the environment and the well-being of their citizens. My research skills are focused on implementing the complexity of complex dynamic systems and networks, crossing behavioral scales from the interaction of human behaviors to the technical networks of the territory. The book "complex systems, smart territories and mobility" from the Springer's Understanding Complex Systems series, which will be published in January 2021 (https://www.springer.com/gp/book/9783030593018) illustrates the research coordination actions that I lead in these fields. '''Slimane Ben Miled, Senior Researcher at Pasteur Institute of Tunis, Professor at ENIT''' Our Tunisian consortium want to constitute a collaborative Research Training Programs to increase data science capacity related to health research in Africa by building trainings and enhancing institutional capacity at African academic institutions. The academy/project is based on 4 pillars to build a training ecosystem for Data and Engineering Science in health. # A platform of federated master’s programs with à la carte optional courses covering informatics/computer science, biomedical informatics, data science, statistics, and public health). Each program will keep its independence, with a mention to the academy label, and this platform will allow to enrich the training with optional modules, seminars, and courses in the partner institutions. New curricula will be created in relation to ethical issues. # Network of Doctoral programs and Executive programs # Platform of federated Business incubator and a career center offers training, support and funding for projects related to the project’s topic. This challenge is in perfect agreement with the Sustainable Development Goal 3 and the CS-DC flagship PHYSIOMES (Personalized Health phYSIcally, sOcially and Mentally for Each in their networkS). '''Masa Funabashi: researcher of open complex systems at Sony Computer Science Laboratories, Inc.''' I would like to contribute to the executive committee of CS-DC on the following two pillars: * Promote the FOOD (From smart agrOecOnomy to smart fooD) flagship project that aims to resolve the health-diet-environment trilemma through the promotion of sustainable food systems, in collaboration with the e-lab "human augmentation of ecosystems" members institution: Sony CSL, Synecoculture Association, CARFS, and those who wish to participate in CS-DC collaboration. * Construct a basic e-learning content on Synecoculture and ecological literacy as a part of CS-DC MOOCs and perform initial trials, principally in ECOWAS countries, through the Sony CSL-CARFS collaboration. Through the development of on-going activities in FOOD project and making synergy with other flagship projects, I would like to contribute CS-DC as a member of the executive committee and realize further extension toward the achievement of global sustainability goals such as SDGs. '''Dr Mohamed Abdellahi (Ould BABAH) Ebbe, Mauritania, ''' * General Director of the Institut du Sahel/CILSS www.insah.org; * Executive Director of the Orthopterist Society (400 researchers among the globe) https://orthsoc.org/ By obtaining the honor of having your hoped-for confidence for this post of member of the executive council of the CS-DC, I will work, in priority and in the short term on two main subjects: # '''The transboundary plague of the Desert Locust (Schistocerca gregaria (Forska l , 1775))''' This plague of the Desert Locust of more than 3000 years that cites all our holy books (the Tourah, the Bible and the Koran) and which continues to be present to this day and to wreak devastating devastation. In case of invasion, it can affect the agriculture and pastures of about 25 countries including those of the poorest countries of the world, from Mauritania to India, while its best and most effective strategy of struggle is preventive struggle by targeting its first centers of gregarization which are very small in space and much better known today. In 2005, the costs of its struggle in the Sahel and North Africa amounted to half a billion dollars, with 8 million farmers and pastoralists affected in the Sahel. It also massively invaded Asia and Africa. 'East Africa in 2020. On this subject, I have spent 30 years studying and fighting and developing a national strategy against this scourge which has made it possible to establish a whole prevention model and an institutional, technical, operational mechanism. and scientific effective in my country that can be adapted and copied and in all other affected countries: Biogeography of the desert locust Schistocerca gregaria, Forskal, 1775: Identification, characterization and originality of a gregarious focus in central Mauritania (HR.HORS COLLEC.) (French Edition) - Babah Ebbe, Mohamed Abdallahi | 9782705670573 | Amazon.com.au | Books http://www.worldbank.org/en/news/feature/2010/01/07/improved-ways-to-prevent-the-desert-locust-in-mauritania-and-the-sahel, http: // whatsnext.blogs.cnn.com/2012/02/02/in-mauritania-sunny-with-a-chance-of-locusts/ # '''The Sahel Institute (INSAH) www.insah.org of the Permanent Interstate Committee for Drought Control (CILSS)''' that I lead and which has been doing extraordinary work for almost half a century in the field of research and development of animal and plant production techniques and also in the field of support for demographic, population and development policies, in favor of the populations of our 13 Sahelian, coastal and island member countries. This work covered the majority of good practice technologies in the field of plant and animal production, natural resource management, land restauration, cultivation techniques, post-harvest, machining, dehulling operations technology. / ginning, Conservation and storage, good resilience practices Research on the demographic dividend, gender and the empowerment of women and the Population / Development interrelations ... etc The results of all this work are contained in a database. data, online http://publications.insah.org/, containing more than 1,500 books, scientific and technical articles that will have to be modernized and connected to the CS Meta data. Finally, I will work actively with my colleagues on the Executive Board on all aspects of other cross-border scourges but also all aspects of improving agro-sylvo-pastoral production tools as well as the fight against poverty and food insecurity and nutrition in line with the goals (SDGs) '''Dr. Habil. László Barna Iantovics, Associate Professor at “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures, Romania.''' With the present manifesto, I would like to candidate as a member of the CS-DC Executive Committee. I am the representative of “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures from Romania in CS-DC. Some of my research and academic activities are related to the complex systems, including: publications, organization of conferences (e.g. Symposium on Understanding Intelligent and Complex Systems - UICS 2009; 1st Int. Conf. on Complexity and Intelligence of the Artificial and Natural Complex Systems Medical Applications of the Complex Systems. Biomedical Computing -CANS 2008; 1st Int. Conf. on Bio-Inspired Computational Methods Used for Difficult Problems Solving. Development of Intelligent and Complex Systems - BICS 2008), contribution to conference committees (e.g. Int. Conf. Emergent Proprieties in Natural and Artificial Complex Systems - EPNACS 2007; Workshop on Complex Systems and Self-organization Modeling -CoSSoM 2009), preparing journal special issues (e.g. Special Issue on Complexity in Sciences and Artificial Intelligence; Special Issue on Understanding Complex Systems), participating in journal’s editorial board (e.g. Complex Adaptive Systems Modeling -CASM, SpringerOpen), and contribution to research in projects and projects coordination (Social network of machines- SOON; Hybrid Medical Complex Systems -ComplexMediSys). I was the director of the center Advanced Computational Technologies – AdvCompTech in the frame of my university. At present, I am the director of the Center for Advanced Research in Information Technology from my university. I would like much deeper involve myself in the life and activities of the CS-DC community. My objectives: * To involve junior and senior researchers from my university in activities regarding research and education. To motivate universities and research institutes from my country to contribute to CS-DC. I consider also universities and research institutes with that I have collaboration in the past. * To PROPOSE the formation of a so-called doctoral and postdoctoral students group. In the case of doctoral and postdoctoral students probably in time more students would like to be involved in activities. In this framework, I suggest the organization yearly 3 times (from 4 to 4 months) workshops in that all the interested students could discuss, present their research and research in progress. With this occasion in the frame of workshops if there is interest could be established separate sessions with presentations also by B.Sc. and M.Sc. students. * To PROPOSE the strengthening of the following research direction with the general topic: intelligent complex systems and machine intelligence measuring. One of the subtopic by interest will be complex systems approaches in medicine and healthcare. To be accomplishable this subject I propose in a first step the formation of a group of interested persons, after then the establishment of the functionality of the group, for example: discussions when are subjects that should be discussed etc. ==Elected members to the Executive Committee & as (Deputy-)Presidents== '''Jeffrey JOHNSON, Professor of Complexity Science and Design, The Open University, UK''' I offer myself as a candidate both to be President and to the Executive Committee of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC) so that I can help to drive it forward to achieve it goals. I am particularly committed to our educational efforts. I have made four MOOCs on the FutureLearn Platform for CS-DC ( https://www.futurelearn.com/partners/unesco-unitwin-complex-systems-digital-campus ): Global Systems Science (2015-16); Systems Thinking and Complexity (2017-18); First Steps in Data Science with Google Analytics (2018-19) and COVID-19 - Pandemics, Modelling and Policy (2020). CS-DC has a great opportunity to become the global university providing interdisciplinary education for a better world. I am also committed to our research mission with UNESCO towards the achieving the U.N. Sustainable Development Goals. My own research on representing the dynamics of complex multilevel systems is relevant to many of the research initiatives of CS-DC. I have extensive experience working within the complex systems community. I have run various coordination actions supporting research programmes funded by the European Commission, I am a founder member and past president of the Complex Systems Society, and I am Deputy-President of the CS-DC. I believe this experience will enable me to make a significant contribution the CS-DC over the next three years. '''Paul BOURGINE, present President of the UNESCO UniTwin CS-DC, Complex Systems Institute of Paris''' I offer myself as a candidate both to be President and to the Executive Committee of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC). If elected, my main commitment is to create the conditions for a self-organized development of our UniTwin UNESCO CS-DC as autonomous communities of communities. This self-similar development will be the case both for the two main branches the UniTwin branch of our institutional members and the global eCampus branches of our individual scientific members: * for the UniTwin branch, the communities of communities are a territorial cascade with Smart Continents, smart countries, smart cities for their sustainable development according our flagship TIMES (Territorial Intelligence for Multilevel Equity and Sustainability). The roadmap is always the same, i.e. the cascade of the 17 Sustainable Development Goals and their 169 Targets: but their relative importance and coherence within this cascade vary from one territory to the others. The institutional members of the UniTwin branch have signed their agreement with the Cooperation Programme signed with UNESCO. In 2021, the CS-DC will ask for a cascade of agreements inside each institutional member, in order to have a "one for all" amplification within the other branch, the e-campus branch. * for the eCampus branch, the cascade of communities is along the refinement cascades when studying the theoretical and experimental challenges of complex systems. With Smart Continents'21, scientists are proposing their individual challenges that enact basic communities and communities of communities within the e-departments. In the "all for one" return, the roadmap of each university is the cascade of roadmaps within the eCampus where the University has at least one member. Furthermore each community can organise a monthly e-seminar or e-session in workshop as well as in CS-DC'21 for recorded advanced introductions. Such advanced introductions can be the basis for curriculum largely shared by the set of Universities having members in the community cascade of the curriculum. This "accelerator of knowledge and knowhow one for all and all for one" will first benefit to the student curriculum through the flagship POEM (Personalized Open Education for the Masses). This accelerator can be extended through the flagship POLE (Personalized Open Lifelong Education) for a lifelong education. This extended accelerator will be open to all, independently of previously achieved academic levels, respectful of the diversity of social and cultural environments and in a higher and higher inclusive way including refugees, migrants and primary people. genders, religions or ways of life. slg9kpzpx3uopg7k8x8ikx2l4cffzi5 0 285380 2806991 2806856 2026-04-29T13:26:10Z Young1lim 21186 /* Applications */ 2806991 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260429.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> e9lcefur9tck3glcwbfafpr3ju1wtqw 1 306051 2807052 2757627 2026-04-29T20:29:58Z Dronebogus 3054149 /* Proposal for rewriting this article */ new section 2807052 wikitext text/x-wiki ==Changes for 2024== The original 2023 guidelines have been revised in 2024 to make it clearer that any genAI-based text must: * clearly and transparently provide genAI acknowledgement in the edit summary, with a link to the chatbot conversation or model and prompt details * be fact-checked * include appropriate peer-reviewed citation -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:24, 5 July 2024 (UTC) ==Literature searching== I find this page useful but sometimes, I still prefer to find my sources through books and medical journals rather than use the help of Gen AI. [[User:Eva U3259916|Eva U3259916]] ([[User talk:Eva U3259916|discuss]] • [[Special:Contributions/Eva U3259916|contribs]]) 05:53, 10 August 2025 (UTC) :{{ping|Eva U3259916}} For a thorough literature search, prioritise search of academic journal databases. Google Scholar provides very useful search. GenAI may be helpful for cross-checking and seeing whether it suggests any key literature that was missed during database searching. However, I would caution against relying on genAI as the primary tool for academic literature searching. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:39, 4 October 2025 (UTC) == Proposal for rewriting this article == The following is my proposed draft In its entirety: '''“Don’t.”''' [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 20:29, 29 April 2026 (UTC) slesl53zqw837pzbohhnhm6u43lcdos 0 307921 2806995 2806929 2026-04-29T13:48:20Z TMorata 860721 shortned sentence with no citation 2806995 wikitext text/x-wiki {{:Global Audiology/Header}} {{:Global Audiology/Oceania/Header}} {{CountryHeader|File:Australia (orthographic projection).svg|https://en.wikipedia.org/wiki/Australia}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Australia Australia], officially the Commonwealth of Australia is a country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. Australia has a population of approximately 27.7 million. Aboriginal and Torres Strait Islander peoples make up approximately 3.2% of the Australian population. Although English is not the official language in law, it is the de facto official and national language. The 2021 Census found that 167 Indigenous languages were spoken at home by Indigenous Australians. {{HTitle|History of Audiology}} Audiology emerged as a profession in Australia during the 1940s. Two major events drove its establishment as a medical specialty. First, many servicemen returned from World War II suffering from hearing loss caused by injuries and noise exposure. Second, the 1940-41 rubella epidemic damaged the hearing of numerous children. These circumstances led Australia to establish the [https://www.nal.gov.au/ National Acoustics Laboratory (NAL)] in 1947. The laboratory began testing hearing and fitting hearing aids for veterans and affected children. Twenty years later, in 1967, NAL expanded its services to include pensioners. A pivotal moment in establishing audiology as a distinct profession came when NAL's original director determined that audiological functions such as hearing testing, hearing aid fitting, and associated rehabilitation should be performed by professionally qualified personnel rather than technicians. The initial debate centered on whether preschool teachers or psychologists would be better suited for these roles. Psychologists were ultimately selected, reportedly because they were considered better equipped to identify malingering among veterans. The first audiologists in Australia were psychologists who undertook postgraduate training in speech therapy and hearing care. Throughout the 1950s and 1960s, the number of practicing audiologists grew steadily and, in the latter half of this period, by the creation of positions in hospital settings. Around 1960, informal gatherings of audiologists began in Sydney and Melbourne. In Melbourne, these typically consisted of monthly lunches followed by afternoon seminars. These regional networking efforts eventually led to the formation of the Audiological Society of Australia in May 1968, which began with ten foundation members representing four states. The majority of audiologists practicing before 1970 came from psychology backgrounds, though some arrived from fields including teaching, physics, and speech pathology. Significant developments after 1970 transformed the profession considerably. The establishment of formal audiology training programs in Brisbane, Melbourne, Sydney, and Perth marked a crucial step toward professional standardization. Audiology subjects were also incorporated into speech-language pathology courses at several institutions. The Australian government and other employers granted official recognition to audiology as a distinct profession, lending credibility and structure to the field. Employment opportunities expanded significantly, particularly in academia and private practice. Audiological research flourished, with notable emphasis on hearing aids and cochlear implants, though research extended into many other areas as well. The scope of audiological services broadened considerably during this period. The profession established regular national conferences beginning in 1974, held biennially, along with other specialized scientific meetings. The Australian Journal of Audiology was founded in 1979, providing a dedicated publication outlet for Australian audiological research. Publications by Australian audiologists in both national and international venues increased substantially, and interactions with international colleagues intensified through conferences, visits, and collaborative research projects. {{HTitle|Incidence and Prevalence of Hearing Loss}} Hearing loss represents a substantial and growing public health burden in Australia. National estimates from the Australian Government Roadmap for Hearing Health indicate that, by 2019, approximately 3.6 million Australians were living with some form of hearing impairment. The prevalence of moderate and more severe hearing loss (≥ 40 dB HL) in children ranges from 1.04 per 1000 live births at 3 years of age to 1.57 per 1000 live births for children between 9 and 16 years of age. Mild hearing loss (< 40 dB HL) increases from 0.28 per 1000 live births at 3 years to 1.68 per 1000 live births at 9 years and older.<ref>{{Cite journal|title=The Ages of Intervention in Regions With and Without Universal Newborn Hearing Screening and Prevalence of Childhood Hearing Impairment in Australia|url=http://www.portico.org/Portico/article?article=pf16jrb3hm|journal=Australian and New Zealand Journal of Audiology|date=2006-11-01|pages=137–150|volume=28|issue=2|doi=10.1375/audi.28.2.137|first=Teresa Y.C|last=Ching|first2=Ron|last2=Oong|first3=Emma van|last3=Wanrooy}}</ref> Among urban Australian school-aged children (5 to 7 years), the prevalence of bilateral hearing loss ≥26 dB was estimated to 2.1% as reprted in 2006.<ref>{{Cite journal|title=Cross-sectional prevalence and risk factors for otitis media and hearing loss in Australian children aged 5 to 7 years: a prospective cohort study|url=https://www.theajo.com/article/view/4259/html|journal=Australian Journal of Otolaryngology|date=2020-03|pages=0–0|volume=3|doi=10.21037/ajo.2020.02.02|first=Christopher G.|last=Brennan-Jones|first2=Hrehan H.|last2=Hakeem|first3=Cheryl Da|last3=Costa|first4=Weijie|last4=Weng|first5=Andrew J. O.|last5=Whitehouse|first6=Sarra E.|last6=Jamieson|first7=Robert H.|last7=Eikelboom}}</ref> In a national study, the prevalence of bilateral and unilateral hearing loss ≥16 dB HL was 9.3% and 13.3%, respectively. Slight losses (16-25 dB HL) were more prevalent than mild or greater losses (≥26 dB HL). <ref>{{Cite journal|title=Cross-sectional epidemiology of hearing loss in Australian children aged 11–12 years old and 25-year secular trends|url=https://adc.bmj.com/lookup/doi/10.1136/archdischild-2017-313505|journal=Archives of Disease in Childhood|date=2018-06|issn=0003-9888|pages=579–585|volume=103|issue=6|doi=10.1136/archdischild-2017-313505|language=en|first=Jing|last=Wang|first2=Carlijn M P|last2=le Clercq|first3=Valerie|last3=Sung|first4=Peter|last4=Carew|first5=Richard S|last5=Liu|first6=Fiona K|last6=Mensah|first7=Rachel A|last7=Burt|first8=Lisa|last8=Gold|first9=Melissa|last9=Wake}}</ref> In the Blue Mountains Hearing Study, 33% of older adults had some degree of hearing loss at baseline, with a 5-year incidence of 17.9%.<ref>{{Cite journal|title=Five-Year Incidence and Progression of Hearing Impairment in an Older Population|url=https://journals.lww.com/00003446-201103000-00010|journal=Ear & Hearing|date=2011-03|issn=0196-0202|pages=251–257|volume=32|issue=2|doi=10.1097/AUD.0b013e3181fc98bd|language=en|first=Paul|last=Mitchell|first2=Bamini|last2=Gopinath|first3=Jie Jin|last3=Wang|first4=Catherine M.|last4=McMahon|first5=Julie|last5=Schneider|first6=Elena|last6=Rochtchina|first7=Stephen R.|last7=Leeder}}</ref> In 2022 hearing loss was estimated to affect 74% of people aged over 70 in Australia.<ref>{{Cite journal|title=Hearing loss, cognition, and risk of neurocognitive disorder: evidence from a longitudinal cohort study of older adult Australians|url=https://www.tandfonline.com/doi/full/10.1080/13825585.2020.1857328|journal=Aging, Neuropsychology, and Cognition|date=2022-01-02|issn=1382-5585|pages=121–138|volume=29|issue=1|doi=10.1080/13825585.2020.1857328|language=en|first=Paul A.|last=Strutt|first2=Amanda J.|last2=Barnier|first3=Greg|last3=Savage|first4=Gabrielle|last4=Picard|first5=Nicole A.|last5=Kochan|first6=Perminder|last6=Sachdev|first7=Brian|last7=Draper|first8=Henry|last8=Brodaty}}</ref> {{HTitle| Audiology Education in Australia}} Audiologists in Australia complete a minimum of five years of university education, including a two-year master’s level audiology program accredited by Audiology Australia. Currently, seven Australian universities offer Audiology Australia–accredited postgraduate audiology programs. Following graduation, Audiology Australia members are required to complete a one-year supervised clinical internship.During this period, interns practice under the supervision of an Audiology Australia Accredited Audiologist, facilitating a structured transition into professional practice and ensuring high standards of service delivery (Audiology Australia, n.d.) Currently, six universities across Australia offer master's programs in audiology. These include Macquarie University, University of Queensland, Flinders University, Melbourne University, La Trobe University, and the University of Western Australia. The University of Western Australia offers a joint master's/PhD in clinical audiology. Australian master's programs require graduates to meet a set of clinical competencies and complete 250 hours of clinical experience before graduation. Graduates must also complete a one-year clinical internship if they want to provide services to clients in the large government-funded sector. During this year, experienced audiologists supervise interns and prepare them for independent clinical practice. After completing the internship, new graduates receive certification from either Audiology Australia or the Australian College of Audiology. Both organizations represent audiologists professionally in Australia. {{HTitle|Scope of Practice and Licensing}} Audiologists need to meet the relevant membership and clinical competency requirements set by Australian Practitioner Professional Bodies to practice in Australia. They need to hold full membership in Audiology Australia with a Certificate of Clinical Practice (CCP) and/or full/ordinary membership in the Australian College of Audiology (ACAud) with Hearing Rehabilitation Specialist (HRS) and Diagnostic Rehabilitation Specialist (DRS) competencies. Audiologists must complete at least the equivalent of an Australian university master's degree in clinical audiology. Audiologists in Australia work with clients of all ages, from infants to older adults, including clients with complex needs. They assess hearing and auditory function, vestibular function, tinnitus, auditory processing function, and neural function. Audiologists perform diagnostic tests, including advanced tests using electrophysiological methods. They provide aural, vestibular, and tinnitus rehabilitation as well as communication training. Audiologists offer a range of rehabilitation services, including counseling and prescribing and fitting various devices and aids. These include bone conduction aids, FM and other remote sensing systems, hearing aids, hearing assistive technology, and earplugs (custom noise/swim/musician plugs). Audiologists possess knowledge of implantable devices such as cochlear implants, middle ear implantable hearing aids, fully implantable hearing aids, and bone anchored hearing aids.They collaborate with other professionals when applying these devices in rehabilitation. In addition to audiologists, hearing services in Australia are also provided by audiometrists. Audiometrists typically complete vocational or industry-based training programs focused on hearing assessment and hearing aid provision. Their scope of practice generally includes conducting hearing tests, fitting and managing hearing aids, and providing basic hearing rehabilitation services, particularly in community and private practice settings. Compared to audiologists, audiometrists usually have a more limited scope of practice, particularly in areas such as complex diagnostic testing, vestibular assessment, and electrophysiological measures. Audiometrists may be affiliated with professional organizations such as the Australian College of Audiology or other industry bodies. {{HTitle|Professional and Regulatory Bodies}} Australia does not have a single statutory licensing authority for audiologists or audiometrists. Instead, hearing care professionals are represented by two main Practitioner Professional Bodies: [https://audiology.asn.au/Audiology Australia] and [https://www.acaud.com.au/Australian College of Audiology incorporating HAASA (ACAud inc. HAASA)]. Audiology Australia is the peak and largest accrediting professional member body for audiologists in Australia, while ACAud inc. HAASA represents both audiologists and audiometrists. Audiologists must meet membership and clinical competency requirements set by Audiology Australia and/or ACAud, typically including completion of a master’s-level degree in clinical audiology and a supervised clinical internship. New members of Audiology Australia become accredited audiologists after completing this internship. Accreditation is valid for one year and must be renewed annually through continuing professional development (CPD). Members are required to demonstrate sufficient professional development over the previous 12 months, which may include conferences, seminars, training courses, and research activities. Audiometrists must meet the relevant membership and competency requirements of ACAud and/or HAASA and typically complete at least a diploma-level Technical and Further Education (TAFE) qualification in audiometry or a bachelor’s degree. These professional bodies define scope of practice, competency standards, and continuing professional development requirements; however, they do not function as statutory regulatory authorities.<ref>{{Cite report|title=Scope of Practice for Audiologists and Audiometrists|url=https://www.acaud.com.au/wp-content/uploads/2019/07/Scope-of-Practice-for-Audiologists-and-Audiometrists.pdf|institution=Audiology Australia; Australian College of Audiology; Hearing Aid Audiometrist Society of Australia|date=2016-09-20}}</ref> '''Code of Conduct''' Audiology Australia members are obliged by a code of conduct. Members must also comply with the Criminal History Policy and Mandatory Declarations Policy.On 12 September 2025, Australia's health ministers agreed to regulate audiology under the National Regulation and Accreditation Scheme. This development requires the Australian Health Practitioner Regulation Agency (AHPRA), which administers NRAS, to provide certification and accreditation to audiologists. {{HTitle|Ongoing audiology research}} Audiology research in Australia are conducted across a wide range of institutions, including universities, independent research institutes, and national organizations. Key contributors include the [https://www.nal.gov.au/National Acoustic Laboratories (NAL)], [https://www.earscience.org.au/ Ear Science Institute Australia], and universities such as the [https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology University of Melbourne], [https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing Macquarie University], University of Queensland, La Trobe University, and Flinders University.<ref>{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}</ref><ref>{{Cite web|title=Ear Science Institute Australia|url=https://www.earscience.org.au/|website=Ear Science Institute Australia|access-date=2026-04-28}}</ref><ref>{{Cite web|title=Audiology and hearing research|url=https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing|website=Macquarie University|access-date=2026-04-28}}</ref><ref>{{Cite web|title=Audiology and Speech Pathology|url=https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology|website=University of Melbourne|access-date=2026-04-28}}</ref>. Research areas include hearing assessment, hearing devices, cochlear implants, tinnitus, vestibular disorders, speech perception, auditory neuroscience, digital health, artificial intelligence, and public-health approaches to hearing care. NAL continues to play a major role through research on adult and paediatric hearing loss, technology, personalized care, and listening difficulties. Its current work includes improving access to hearing services, developing targeted solutions for everyday listening difficulties, and exploring the use of artificial intelligence to improve hearing screening, diagnosis, management, and ongoing support.<ref>{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}</ref> A major national priority is improving hearing health equity. Australian researchers and service providers are working to reduce the impact of otitis media-related hearing loss among Aboriginal and Torres Strait Islander children, often through community-informed and collaborative approaches. Australia also has a national Hearing Services Program that provides subsidised hearing services and devices to eligible Australians with hearing loss. Eligibility includes children and young adults under 26, Aboriginal and Torres Strait Islander people who meet program criteria, and many older adults who qualify through pensioner or veteran status.<ref>{{Cite web|title=Hearing Services Program|url=https://www.health.gov.au/our-work/hearing-services-program|website=Australian Government Department of Health, Disability and Ageing|access-date=2026-04-28}}</ref> {{HTitle|Challenges, Opportunities and Notes}} Australia's audiology sector struggles with several interconnected problems, including insufficient numbers of hearing specialists in regional areas, difficulties accessing services across vast distances, requirements for culturally appropriate care, hearing loss going undetected in elderly care settings, and limited uptake of remote audiology services, even though strong government support programs exist. Primary concerns involve uneven distribution of audiologists across the country, absence of a unified nationwide approach, social stigma surrounding hearing loss, and technological barriers, all of which restrict healthcare access for people living in rural and remote locations, Indigenous Australians, and senior citizens, highlighting the need for enhanced professional training and improved digital health systems. <ref>{{Cite journal|last=Mui|first=Boaz|last2=Lawless|first2=Michael|last3=Timmer|first3=Barbra H. B.|last4=Gopinath|first4=Bamini|last5=Tang|first5=Diana|last6=Venning|first6=Anthony|last7=May|first7=David|last8=Muzaffar|first8=Jameel|last9=Bidargaddi|first9=Niranjan|date=2025-01-02|title=Australian hearing healthcare stakeholders’ experiences of and attitudes towards teleaudiology uptake: a qualitative study|url=https://www.tandfonline.com/doi/full/10.1080/2050571X.2024.2372171|journal=Speech, Language and Hearing|language=en|volume=28|issue=1|doi=10.1080/2050571X.2024.2372171|issn=2050-571X}}</ref> ''Workforce & Access Issues:'' Regional shortages: Audiologists concentrate in cities, leaving rural/remote areas under-serviced Geographic barriers: Vast distances make consistent care difficult for remote communities Aged care gaps: Hearing loss frequently missed due to lack of staff training and awareness.<ref>{{Cite journal|last=El-Saifi|first=Najwan|last2=Campbell|first2=Megan E.J.|last3=George|first3=Neha|last4=Keay|first4=Lisa|last5=Kumaran|first5=Sheela|last6=Meyer|first6=Carly|last7=Miller Amberber|first7=Amanda|last8=Newall|first8=John|last9=Dawes|first9=Piers|date=2025-09-05|title=Barriers and enablers to hearing service provision in aged care settings in Australia: perspectives from hearing clinicians|url=https://www.tandfonline.com/doi/full/10.1080/14992027.2025.2554236|journal=International Journal of Audiology|language=en|pages=1–12|doi=10.1080/14992027.2025.2554236|issn=1499-2027}}</ref> Poor integration: Limited connection between audiology and other allied health services in aged care. ''Service Delivery & Technology Problems:'' Tele-audiology underutilized: Barriers include poor infrastructure, restrictive policies, inadequate funding, and limited clinician training Digital literacy gaps: Patients, especially elderly, struggle with comfort and skills for remote care Digital therapeutics challenges: New technologies need stronger evidence bases before widespread adoption Technology implementation hurdles: Clinicians lack confidence in integrating new digital tools. ''Patient & Cultural Barriers:'' Stigma: Hearing loss viewed as "invisible disability," causing delayed treatment Low public awareness: Limited understanding of hearing loss impacts and treatment options Multicultural needs: Services must be culturally sensitive and linguistically appropriate {{HTitle|Audiology Service Providers and Advocacy Groups}} Australia hosts a robust network of organizations dedicated to supporting individuals with hearing loss, ranging from frontline service providers to advocacy groups and research institutions. ''Leading Service Providers'': '''NextSense''' stands as one of Australia's premier not-for-profit organizations addressing both hearing and vision loss. The organization delivers comprehensive clinical services, conducts research, and operates Australia's largest cochlear implant program. Beyond direct services, NextSense provides educational support and therapeutic interventions for children and adults navigating sensory loss. '''Hearing Australia''' functions as a government-funded authority that delivers subsidized hearing services to eligible Australians, including pensioners, veterans, Aboriginal and Torres Strait Islander people, and young Australians under 26. It serves as the primary hearing healthcare provider across the country, operating clinics nationwide and supplying hearing devices to those who qualify for government support. ''National Advocacy and Peak Bodies'': [https://www.health.gov.au/topics/ear-health-and-hearing/support-services Deafness Forum Australia] operates as the national peak body representing the interests of all Australians with hearing loss. The organization advocates for policy changes, promotes hearing health awareness, and connects individuals with resources across the spectrum of hearing impairment. Deafness Forum Australia plays a crucial role in ensuring that hearing loss remains visible in national health discussions and that the needs of the deaf and hard-of-hearing community influence government policy. '''Audiology Australia (AudA)''' serves as the professional association for audiologists, establishing clinical standards and ethical guidelines for hearing healthcare practitioners. Although not a charity, AudA maintains partnerships with various hearing organizations and provides the public with access to qualified audiologists through its member directory. {{HTitle|References}} {{reflist}} ==External Links== * https://audiology.asn.au/ * https://www.audiology.org/news-and-publications/audiology-today/articles/a-hearing-report-from-australia/ * https://audiology.asn.au/standards-guidelines/scope-of-practice/ * https://www.auditdata.com/insights/cases/enhancing-audiology-care-in-australia-territory-hearings-success-with-manage-software * https://www.health.gov.au/topics/ear-health-and-hearing/support-services * https://www.acnc.gov.au/charity/charities/f6db5b5d-3aaf-e811-a963-000d3ad24077/profile {{Global_Audiology Authors |name1=Biraj Bhattarai |name2=Sajana Aryal |role1=Contributor |role2=Contributor |linkedin1=https://www.linkedin.com/in/biraj-bhattarai-3172931a3 |linkedin2=https://www.linkedin.com/in/sajana-aryal-209612187/ }} [[Category:Audiology]] [[Category:Australia]] 9h2nmwdlswi00lsdhf97plxkgvkp0s6 0 317041 2807082 2766830 2026-04-30T03:59:17Z DavidMCEddy 218607 /* Bibliography */ syntax 2807082 wikitext text/x-wiki :''This is a discussion of a Zoom interview to be recorded 2024-12-13 with communications professor [[w:Victor Pickard (professor)|Victor Pickard]] about his research discussing how information is a public good and the public policy implications of that claim. A 29:00 mm:ss podcast excerpted from the companion video will be posted here after it is released to the fortnightly "Media & Democracy" show''<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Information is a public good per communications prof Pickard.webm|thumb|Interview claiming that information is a public good and discussing market failures in for-profit media according to [[w:Victor Pickard (professor)|Victor Pickard]], communications professor in the [[w:Annenberg School for Communication at the University of Pennsylvania|Annenberg School for Communication at the University of Pennsylvania]]]] [[File:Information is a public good per communications prof Pickard.ogg|thumb|29:00 mm:ss podcast from Interview conducted 2024-12-13 regarding professor Pickard's claim that information is a public good and public policy implications of that claim.]] Professor [[w:Victor Pickard (professor)|Victor Pickard]] in the [[w:Annenberg School for Communication at the University of Pennsylvania|Annenberg School for Communication at the University of Pennsylvania]] discusses how information is a public good and public policy implications of that claim. He is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> == Primary concerns == It is in your best interests and mine to help supporters of our worst enemies get information they want, because doing so will make it harder for their leaders and ours to convince us to support policies that may threaten our lives and futures to please those who control most of the money for the media. Research suggests that better media reduces political corruption and improves the quality of life for the vast majority. News deserts, ghost newspapers, and major media conglomerates have the opposite effect, encouraging public officials to focus less on protecting the interests of voters and often clandestinely reward campaign contributors to the detriment of the electorate. Commercial media are not likely to expose this corruption, because they make money selling advertising to the beneficiaries of that political corruption and from increasing political polarization and violence.<ref>Pickard (2020, 2023). See also [[Information is a public good: Designing experiments to improve government]].</ref> If we look at how the major media in the US are generally funded, "Their business model ... at least for about 125 years or so has been advertising. ... This really developed somewhere in the mid to late 1800s".<ref>This interview also briefly mentioned John and Silberstein-Loeb, ed (2015). ''Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet'', cited by Pickard (2020), which places these changes in a much broader context. McChesney and Nichols have suggested that that most people alive today benefit from subsidies for newspapers in the US in the early 1800s, even though they've never read those newspapers. This follows, because those newspapers encouraged literacy and limited political corruption, both of which helped the new US stay together and grow both in land area and economically, while contemporary New Spain / Mexico fractured, shrank, and stagnated economically. For more on this, see [[The Great American Paradox]]. People in other countries benefit from scientific advances that would not have occurred without that diverse media environment in the US before media consolidation began in the late 1800s.</ref> The newspaper industry, "even in its beleaguered state, is still the source of most of our original reporting, original news and information that gets disseminated. So newspapers have historically been sort of the information feeders for our entire media ecosystem. ... But actually, media subsidies are as American as apple pie. Going back to our first major communication system, which was the postal system, and our newspapers were tremendously subsidized." "Then there was this transformation of the logic driving our newspaper industry, ... this primary business model was to deliver audiences to advertisers. ... That began to come apart in the early 2000s, when readers and advertisers migrated to the web ... . There is no viable economic model to support the level of journalism that democracy requires. We have to start thinking about other models ... ." Earlier this year, Pickard published an article with Neff, which compared newspapers in 33 different countries.<ref>Neff and Pickard (2024).</ref> "In a kind of comparative framework ... we are literally off the chart for how little we fund our public media. ... At a national level it comes to ... a little bit over a $1.50 per person per year that we pay at the Federal level towards our public media. If you throw in local and regional and state subsidies, it gets up to a little bit over $3 per person per year. Now compare that to the Brits, who spend about $100 per person per year for the BBC. Or look at northern European countries where they're spending far more than that." Conservative organizations that evaluate the level of democracy have found that "the strongest democracies on the planet ... also happen to have the strongest public media systems on the planet. ... These same institutions have qualified the US as being a flawed democracy. We've been considered a flawed democracy for a number of years now. And, of course, we have a very weakly funded public broadcasting system. So what this shows at the very least, is that if you publicly subsidize your media systems, your public media systems, if you make those public investments in the news and information that democracy requires, these countries are not sliding into totalitarianism. They're not becoming fascist countries overnight. Quite the contrary. They're they're very strong. There are shining exemplars of democratic countries. This doesn't mean that we shouldn't also be concerned about state capture of public media systems, and we can point to some cautionary tales like in Turkey and Hungary and Poland, you know, that can happen. But those are the exceptions. Most of these strong democracies have strong public broadcasting systems, public media systems. So I would argue that that should also be part of our redemocratization project here in the United States is to actually fund our public media so that they don't have to rely on private funders. NPR gets more than a third of its money from corporate funding, which sort of defeats the purpose.. It's a misnomer even to call it public broadcasting if they're taking in all this corporate money, and any casual listener or viewer of NPR and PBS will have to sit through what's uphemistically called enhanced underwriting. ... That's kind of absurd for a public media system. So we need to change that. But I do think that that's something we need to focus on more. We need to really build out our public media systems so that it can serve local information needs." McChesney and Nichols (2021, 2022) recommend distributing 0.15% of national income (Gross Domestic Product, GDP) to local news nonprofits on the basis of local elections. Pickard likes their model but prefers other alternatives, like local news bureaus or multimedia centers managed by local elected individuals or selected at random, similar to jury duty. The main point is to provide public funding with a firewall to prevent interference in the content by other government bureaucrats or corporate interests. Pickard continues, "We basically want a system that allows journalists to be journalists, to do the work that originally drew them to the craft ... . Profit a driven media is always going to privilege profits over democracy." == About Pickard == Pickard is a media studies scholar and a professor at the [[w:Annenberg School for Communication at the University of Pennsylvania|Annenberg School for Communication at the University of Pennsylvania]]. He works on the intersections of US and global media activism and politics and the role of the media in political economy.<ref>[[w:Victor Pickard (professor)|Victor Pickard]].</ref> He is also the Chair of the Board of Free Press. He has written or edited six books,<ref><!--Free Press Board-->{{cite Q|Q131398406}}</ref> including (2015) ''America's Battle for Media Democracy'',<ref>Pickard (2015)</ref> and (2020) ''Democracy Without Journalism? Confronting the Misinformation Society''.<ref>Pickard (2020).</ref> == The threat == Internet company executives have knowingly increased political polarization and violence including the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise might have reduced their profits. Documentation of this is summarized in [[:Category:Media reform to improve democracy]]. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Richard R. John and Jonathan Silberstein-Loeb (2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet, Oxford U. Pr.-->{{cite Q|Q131468166}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Victor Pickard (2023) Another Media System is Possible: Ripping Open the Overton Window, from Platforms to Public Broadcasting, Janost-->{{cite Q|Q131398460}} * <!--Victor Pickard (2020) Democracy without journalism? : confronting the misinformation society, Oxford U. Pr.-->{{cite Q|Q131398359}} * <!--Victor Pickard (2015) America's Battle for Media Democracy, Cambridge U. Pr.-->{{cite Q|Q131398237}} [[Category:Politics]] [[Category:News]] [[Category:Media reform to improve democracy]] ouagc4fvr1nvqdd80e2wr8vwlv4nzdw 2 320737 2807059 2805653 2026-04-29T21:07:48Z Tommy Kronkvist 31941 User statistics. 2807059 wikitext text/x-wiki <div style="margin: 0 0 1em 0;">{{userpage}}</div> {{Userboxtop|toptext=Babel:}} {{#babel:sv|en-4|de-2|la-1}} {{Userboxbottom}} [[File:Sorbus torminalis Trunk and canopy.jpg|thumb|300px|The canopy of a Checker tree <small>(''Torminalis glaberrima'')</small>]]<br /> Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,<small>^{[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&limit=1&username=Tommy_Kronkvist (verify)]}</small> to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,<small>^{(<span class="plainlinks">[https://se.wikimedia.org/w/index.php?title=Special:Användare&limit=1&username=Tommy_Kronkvist verify]</span>)}</small> and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].<small>^{(<span class="plainlinks">[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&limit=1&username=Tommy_Kronkvist verify]</span>)}</small> So far (April 29, 2026), I've made just over 391,600 edits to 153 of the Wikimedia sister projects&nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]]. Swedish is my mother tongue&nbsp;– even though I was born in Finland&nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden. I'm only the fourth generation named "Kronkvist". My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning "Matthew's Farm", dating back to at least 1637. {{Clear}} {{User committed identity|a6edd6d2fdbf82621f0cda4e5525c71f8da9b5dfd308242c3c63365e998c32c5406b75448380903265a5403edffd1a0435b61ac943f3c65870db9250f8b884a9|SHA-512|background=#e0e8ff|border=e0e8ff}} dfsoimusetkz2ty9iem1wb8srz89gr2 0 325449 2807067 2781949 2026-04-30T02:41:07Z DavidMCEddy 218607 /* News Deserts */ typo 2807067 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by increasing… increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly, 13 hundredths of a percent of the 2019 US GDP. I mentioned this in my July 17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. Perhaps the most spectacular example that I know of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999 . Roughly a decade later the city was nearly bankrupt in spite of having property taxes out of sight. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly, well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2019) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> lrhpf5m69130laz32wgatgm5fep4xxy 2807069 2807067 2026-04-30T03:23:46Z DavidMCEddy 218607 /* News Deserts */ typo 2807069 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly, 13 hundredths of a percent of the 2019 US GDP. I mentioned this in my July 17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. Perhaps the most spectacular example that I know of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999 . Roughly a decade later the city was nearly bankrupt in spite of having property taxes out of sight. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly, well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2019) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> ct8tr1ii5381tkbsnplwvenm343ov1f 2807070 2807069 2026-04-30T03:24:14Z DavidMCEddy 218607 /* News Deserts */ punct 2807070 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. I mentioned this in my July 17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. Perhaps the most spectacular example that I know of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999 . Roughly a decade later the city was nearly bankrupt in spite of having property taxes out of sight. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly, well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2019) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> 2zfl4kz0vyuwh9kwhbregepwe279pwi 2807072 2807070 2026-04-30T03:27:25Z DavidMCEddy 218607 /* News Deserts */ wdsmth 2807072 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999 . Roughly a decade later the city was nearly bankrupt in spite of having property taxes out of sight. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly, well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2019) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> pwjz763hzqeh13qwcxode6c4kj50a69 2807073 2807072 2026-04-30T03:29:08Z DavidMCEddy 218607 /* News Deserts */ wdsmth 2807073 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999. Roughly a decade later the city was nearly bankrupt in spite of having property taxes out of sight. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2019) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> ietz6p5oqjz13w60bq63tskef8z1r8t 2807074 2807073 2026-04-30T03:30:58Z DavidMCEddy 218607 /* News Deserts */ wdsmth 2807074 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999. Roughly a decade later the city was nearly bankrupt in spite of having property tax rates among the highest in the nation. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2019) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> qlaegdqhaggg6peni4v0bydltbynm5v 2807079 2807074 2026-04-30T03:45:53Z DavidMCEddy 218607 /* Bibliography */ wdsmth 2807079 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999. Roughly a decade later the city was nearly bankrupt in spite of having property tax rates among the highest in the nation. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2018) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> c2uzlfsdx5lgdlcjaoyb3ukqdivjytm 2807080 2807079 2026-04-30T03:51:25Z DavidMCEddy 218607 /* Bibliography */ add Benton (2019) 2807080 wikitext text/x-wiki :''This summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> [[w:List of Pacifica Radio stations and affiliates|Network]].''<ref name=PacificaList><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Media & Democracy lessons for the future.webm|thumb|2025-11-06 summary of research and interviews on media and democracy.]] [[File:Media & Democracy lessons for the future.ogg|thumb|29-minute podcast recorded 2025-11-06 summarizing media & democracy lessons for the future]] [[File:Slides for a discussion of media and democracy.pdf|thumb|Slides summarizing the fortnightly Media & Democracy series and related research]] Spencer Graves<ref name=sg><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> summarizes lessons for the future from the first 37 episodes of the [[:Category:Media reform to improve democracy|Media & Democracy]] series,<ref name=M&D/> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref name=Pacifica/> [[w:List of Pacifica Radio stations and affiliates|Network]].<ref name=PacificaList/> Twenty-nine-minute podcasts of all episodes and videos of most are available with the descriptions of each episode. The series has been fortnightly since 2024-07-30. == Approximate transcript of the video == Welcome to Media and Democracy. I'm Spencer Graves with [[w:KKFI|KKFI]], [[w:Kansas City metropolitan area|Kansas City]] [[w:Community radio|Community Radio]]. You are about to hear a summary of what I think have been the most important things covered in (a) the 37 episodes of ''Media & Democracy'' that I have produced so far and (b) over a decade of studying the research literature on this issue before that. The views you are about to hear are mine and of the sources that I cite, and not of this radio station. If the research summarized herein is replicable as described, then we can reverse the current trend towards increasing political polarization and violence and make progress much easier on virtually every other major issue facing humanity today. A key observation that makes this possible is that progress on every substantive issue that I have studied is blocked because every countermeasure threatens someone with substantive control over the money or the medium. === The media and human cognition === To understand the role of the media in political economy, I feel a need to mention key results in human cognition. First, * Everything we think we know is coded in systems of connections between neurons in our brains. * These systems are more unique than fingerprints, and evolve over time. * The words we use do not mean the same thing - to two different humans, - nor to the same human at two different points in time. We can overcome differences of opinion and war and build a better world for all by teaching ourselves how to * talk * politics, * calmly, * with respect and humility, * in a friendly, supportive manner, * with others with whom we may vehemently disagree, because the alternative may be killing people over misunderstandings. This understanding, I believe, can help all of us understand how others believe different from us, because they find different media credible. We can overcome some of these barriers by accepting others where they are and by trying to engage them in non-threatening civil discourse.<ref>Graves (2025).</ref> === Behavioral economics === Virtually everyone thinks they know more than they do.<ref>Research on this is summarized in the Wikipedia articles on [[w:Overconfidence effect|Overconfidence effect]] and the [[w:Dunning–Kruger effect|Dunning–Kruger effect]]. The latter documents the difference between self perception and actual performance: The self perceptions of low achievers tend to be much higher than their actual performance, and this difference is smaller for high achievers and is slightly reversed for some tasks, though not by much.</ref> This is a key result in a relatively new field called [[w:Behavioral economics|behavioral economics]] in the intersection between [[w:Human behavior|human behavior]] and [[w:Economics|economics]]. [[w:Daniel Kahneman| Daniel Kahneman]] won the 2002 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]], even though he was not an economist: He was a research psychologist. He won the 2002 Nobel Memorial Prize in Economics for his leadership in developing this new subfield in the intersection between the Human Behavior and Economics.<ref>Nobel Prize Committee (2002).</ref> Major media exploit this feature of human cognition to please those who control most of the money for the media. Over two years ago, I published an article on Wikiversity on "[[Information is a public good: Designing experiments to improve government]], summarizing the research that I had found to that point on the role of the media in political economy, and recommending research -- experiments -- to quantify the extent to which those research results can actually be replicated. Those concerns about the media also led me to become the primary content producer for Radio Active Magazine,<ref><!--Radio Active Magazine-->{{cite Q|Q57451712}}</ref> a weekly half-hour, magazine-style radio program on [[w:KKFI|KKFI]] about [[w:Activism|activists]]. Since July 30 of last year, Radio Active Magazine has been alternating between local content and national and international experts on the increase in political polarization and violence, and what they think should be done about it. The episodes featuring experts are also distributed as the fortnightly ''Media & Democracy'' series syndicated for the Pacifica Radio Network and made available on Wikiversity under [[:Category:Media reform to improve democracy]], which supports moderated discussions of the issues raised in each episode. The most important thing I think I have gotten from all this work is solid documentation of the value of ''accountability journalism'' relative to ''[[w:access journalism|access journalism]]'': * ''Accountability journalism'' is disseminating information that people with power do not want known. * ''Access journalism'', by contrast, is giving people with power access to an audience to disseminate information they do want known. On June 12 earlier this year, I interviewed [[How news impacts democracy per USD Communications Professor Nik Usher|University of San Diego journalism professor Nik Usher]]. With a co-author, they tallied all of the federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019. They found on average 1.4 more prosecutions for political corruption per year per member of [[w:Institute for Nonprofit News|INN]] in each [[w:Federal judiciary of the United States|US federal court district]] the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3, between 60 and 70 percent, with no statistically significant impact on federal prosecutions for political corruption.<ref>Usher and Kim-Leffingwell (2022).</ref> You and I benefit, we all benefit, from accountability journalism that we have never read nor even heard of as long as enough others got those reports and took effective action to limit malfeasance. Watchdogs protect the people who feed them. For-profit media protect the major corporations, who are the only people who really count in the United States of America today.<ref>There is a long international tradition on "[[w:Corporate personhood|Corporate personhood]]. In the US, this dates from "A headnote issued by the court reporter in the 1886 Supreme Court case ''[[w:Santa Clara County v. Southern Pacific Railroad Co.|Santa Clara County v. Southern Pacific Railroad Co.]]''. It has since been expanded many times in both statutes and Supreme Court decisions. For example, the [[w:Patriot Act|Patriot Act]] of 2001 makes "[[w:Providing material support for terrorism|Providing material support for terrorism]] a felony punishable by life in prison, "if the death of any person results," where 'the term “person” means any individual or entity capable of holding a legal or beneficial interest in property'. Under ''[[w:Holder v. Humanitarian Law Project|Holder v. Humanitarian Law Project]]'', it is a felony to teach nonviolence to anyone designated as supporting a foreign terrorist organization. Apparently, if the US State Department claims that one business ceased operations as a result of the activities of such a foreign terrorist organization, you can get life in prison for teaching nonviolence to anyone the State Department claims supports said organization -- even if you do not know that the human(s) to whom you taught nonviolence are so designated by the State Department. More well-known is the 2010 Supreme Court decision in ''[[w:Citizens United v. FEC|Citizens United]]'', which confirms that corporations are "people" and money is speech.</ref> Many local news outlets funded by advertisers are cheerleaders for local developers, according to [[Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and researcher Dan Hind, whom I interviewed just over a month ago]]. [[w:Community radio|community radio]] stations protect their supporters and their communities and might benefit from the research that I have studied. I feel a need to describe examples. === Example: George Santos === [[w:George Santos| George Santos]] is a former representative of the [[w:United States House of Representatives|US House of Representatives]]. He was forced to resign in 2023 after numerous reports of questionable claims he had made to get elected. In 2024, he was convicted of identity theft and wire fraud in the [[w:United States District Court for the Eastern District of New York|Eastern District of New York]], which is a leader among US federal court districts for having the most members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]], or INN. === Stadium subsidies === Another example: In 2024, April 2, voters in Kansas City rejected, money for sports stadiums.<ref>Lieb (2024).</ref> A few months earlier, I had alerted ''[[w:The Beacon (Kansas City)|The Kansas City Beacon]]'', a local member of INN, to the Wikipedia article on [[w:Stadium subsidy|stadium subsidy]], which cites serious research documenting the lack of benefits for the local economy from stadium subsidies. ''The Beacon'' published stories on that, which were picked up by other news outlets. I believe the coverage in the beacon probably contributed to the electoral defeat of that stadium subsidy -- and to improving the general welfare of the bottom 99% of the Kansas City population. In general, access journalism is cheap as long as you never contradict any leading establishment figure nor ask questions they do not want to answer. ''Accountability journalism'' is expensive: It costs money to check facts, and for-profit media know they could lose money by offending a major advertiser, even if they had all their facts straight. However, if one news agency does it, like a member of INN, for-profit media are often forced to carry it or lose audience. Every media organization sells changes in audience behaviors to the people who give them money. If they lose their audience, they have nothing to sell. === ''The Kansas City Defender'' === Another example: ''[[w:The Kansas City Defender|The Kansas City Defender]]'': Kansas City has two members of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]]: ''The Beacon'' and ''The Kansas City Defender''. The latter is an online news outlet founded by a young [[w:African American|African American]].<ref>In addition to ''The Beacon'' and ''The Kansas City Defender'', the <!--Kansas City Northeast News-->{{cite Q|Q55667687}} is a nonprofit serving Northeast Kansas City. However, as of 2025-11-11, they are not listed as a member of INN and are not listed in the <!--Local Journalism Directory-->{{cite Q|Q136763718}}, maintained by the <!--Media and Democracy Project-->{{cite Q|Q136327862}}. The ''Northeast News'' only became a nonprofit in May of 2022. Their website says they comply with INN's membership standards.</ref> Roughly 3 years ago, ''The Kansas City Defender'' published a report about missing women along the [[w:Prospect Avenue (Kansas City, Missouri)|Prospect Corridor]], a major thoroughfare in a traditionally red-lined African American part of Kansas City, Missouri. ''The Defender'' was vigorously denounced for irresponsible journalism by the chief of police, other local public officials, and news outlets across the nation. Two weeks later, a woman dressed in trash bags was running down the street, knocking on doors, crying for help. An hour later, the owner of the house in which she had been imprisoned was arrested and charged with kidnapping and raping multiple women. In 2024, a body was found, and he was charged with murder. Without the Kansas City Defender, that poor woman might still have escaped, and that serial rapist and murderer might still have been arrested. But ''The Defender'' forced law enforcement and politicians to consider more seriously the charges of poor police protection in that part of Kansas City. Now let me talk about news deserts. === News Deserts === There's a growing body of research describing what happens when local newspapers die. A 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2019>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999. Roughly a decade later the city was nearly bankrupt in spite of having property tax rates among the highest in the nation. The city manager, it turns out, had a compensation package worth $1.5 million a year, well over double that of the President of the United States. And other senior city officials were similarly well-remunerated. Some of the city officials went to jail over that. But it need not have happened if they had a local newspaper.<ref>There is also a growing body of research on the threats from loss of local newspapers: Malfeasance also increases in business as pollution and workplace accidents increase and the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". By contrast, Neff and Pickard (2024) document that the world's leading democracies devote between 0.05 and 0.25 percent of GDP to government funding for media.</ref> Adjusted for inflation, 0.13 percent of GDP is roughly $120 per human per year today. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. Other research documents that when a local newspaper dies pollution, workplace accidents, and cost of capital increases, and innovations declined for local businesses. When innovation declines, economic growth also tends to decline, jeopardizing the future of the nation. Nonprofit executive compensation also increases, so less of what you donate goes to the charitable purpose for which you donated, as documented in my [[Local newspapers limit malfeasance|February 25 interview with Arizona State Accounting Professor Richard White]]. Similarly, voter participation declines, and the far right does better, as documented in my [[News from Germany 1900-1945 and implications for today|June 8th interview with University of British Columbia professor Heidi Tworek]], author of ''News from Germany, the competition to control world communications, 1900-1945''. === Fox === Let me talk about [[w:Fox Broadcasting Company|Fox]]. My interview with City University of New York journalism professor Reece Peck last August 1 documents how during the [[w:Great Recession|Great Recession]] that began in 2007, Fox convinced many of its audience that President [[w:Franklin D. Roosevelt|Franklin Roosevelt]]'s [[w:New Deal|New Deal]] actually made the [[w:Great Depression|Great Depression]] worse, not better. That message helped prevent the US Congress from bailing out the fraud victims of that financial crisis. However, Congress agreed that money had to be injected back into the economy to avoid a repeat of the Great Depression. So major banks that were too big to fail before the crisis were even bigger afterwards, and many financial executives that had created that crisis, many of whom should probably have been prosecuted for fraud instead got multi-million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> === US GDP per capita, Franklin Roosevelt, and war === Let me talk about average annual income, GDP per capita. [[US Gross Domestic Product (GDP) per capita|I've analyzed data on that from 1790 to 2024]]. From 1790 to 1929, the GDP per capita increased at a rate of average about 1.5 percent per year. Then it fell like a rock at over 8% for the four years of the [[w:Presidency of Herbert Hoover|Herbert Hoover administration]]. Then it took off like a rocket at roughly 8% per year during the 12 years of the [[w:Presidency of Franklin D. Roosevelt|Franklin Roosevelt administration]]. Since the end of World War II, after a post-World War II recession that lasted only a couple of years, the US economy has grown on average 2% per year. The data I found on the Franklin Roosevelt administration said that Roosevelt administration actually spent the money required to put humans back to work, and then win World War II with the highest effective tax rates on the ultra-wealthy in US history and with [[w:Office of Price Administration|wage and price controls during World War II]], which largely eliminated price gouging by major businesses that had generated inflation and stifled economic growth in previous wars. During that period, the US had by far the highest rate of increase in average annual income, GDP per capita, adjusted for inflation, of any comparable period in US history, before or since, with only nominal inflation. And inequality also fell dramatically during that period, and only started to increase again around the time that [[w:Ronald Reagan|Ronald Reagan]] became president. By comparison, there have been 3 other major wars in US history. The [[w:War of 1812|War of 1812]], the [[w:American Civil War|Civil War]], and [[w:World War I|World War I]]. All had substantial inflation with economic growth that did not differ substantially from the 1.5% per year that lifted the US from a little over $1,000 per human per year in 1790, adjusted for inflation, to the US's relatively dominant position in the international political economy. The Franklin Roosevelt administration, by contrast, averaged 6% per year in GDP per capita growth between 1933 and 1939, and over 10% during World War II. The US has averaged roughly 2% per year since then, showing how incredibly different the Franklin Roosevelt administration was from the rest of US history. The special circumstances of the Great Depression and World War II gave Franklin Roosevelt the political support needed to spend the money to put people back to work, to end this Depression, and then to win the Second World War. This suggests to me that ''we can do this again'': We only need media that helps convince more humans that it is possible, and that indeed we need to tax the ultra-wealthy in proportion to the benefits they receive from government and do other things to prevent price gouging by major corporations. The major corporate media are watchdogs protecting the people who feed them. And they do not want you know about things like this. === Two primary recommendations === I have two primary recommendations. First, we need citizen-directed subsidies for local news nonprofits with a firewall to prevent political interference in the content, supporting organizations like community radio and members of the Institute for Nonprofit News.<ref>There is a body of evidence that says that most humans trust local news more than non-local sources. See the discussion of news deserts above.</ref> Second, we need to migrate to non-commercial social media like Blue Sky, Mastodon, and PeerTube that do not make money amplifying political polarization and violence.<ref>See the discussion below of the interview with Facebook whistlblower Frances Haugen.</ref> Regarding the first, [[The Great American Paradox|the US Postal Service Act of 1792 provided postal subsidies enacted by the second US Congress and signed by President Washington]] during his first term arguably made major contributions to the long-term sustained growth in the US economy, which I mentioned earlier. Under that act newspapers were delivered up to 100 miles for a penny when first-class postage was between 6 and 25 cents. McChesney and Nichols estimated that in 1840 those subsidies were roughly 0.21 percent of GDP.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> That's roughly $64 billion in today's money, or $190 per human per year. As a result of that act the US had more independent newspaper publishers per million population in the first half of the 19th century than probably at any other time or place in human history.<ref>The claim that the US led the world in independent newspaper publishers in discussed in "[[Media concentration per Columbia History Professor Richard John]]" and John (1995), in particular. [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States of America in 1831, wrote, “There is scarcely a hamlet that does not have its own newspaper.” See Tocqueville (1835, p. 93).</ref> That diversity of newspapers encouraged literacy and limited political corruption and created a political culture that I believe has been a major driver in the economic growth that has given the US its current leadership position in the international political economy. It helped the US stay together and grow, while other countries like contemporary New Spain, then Mexico, fractured, shrank, and stagnated economically.<ref>That diversity of newspaper publishers began to shrink in the 1850s with technology changes that increased the capital required to start a newspaper (John and Silberstein-Loeb, 2015, p. 80). That was followed by consolidation of ownership of newspapers led by [[w:William Randolph Hearst|William Randolph Hearst]]. The introduction of broadcasting made consolidation of ownership easier; John and Silberstein-Loeb (2015). See also Wikiversity, “[[Information is a public good: Designing experiments to improve government]]” and “[[:Category:Media reform to improve democracy]]“. [[:Category:Media reform to improve democracy|That consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself, as documented with the Wikiversity article on "[[Evolution of political polarization in the US Congress]]" (accessed 2025-11-11), which contains plots of data on the evolution of political polarization in the US Congress 1879-2023.</ref> McChesney and Nichols recommended an internet-savvy reincarnation of the newspaper subsidies that the US had 200 years ago. They recommended distributing 0.15 percent of GDP to local news nonprofits via local elections to provide a firewall to prevent political interference in the content.<ref>McChesney and Nichols (2021, 2022).</ref> Many municipalities can raise that kind of money by committing roughly 3% of their budget to subsidize local news nonprofits with a firewall that effectively prevents, as I said, political interference in the content.<ref>Roughly 1 percent of the US workforce are accountants and auditors. Roughly 2 percent of GDP is devoted to advertising. If local governments are comparable to the overall economy,accounting, advertising, and public relations may easily exceed 3% of their budget. More on this appears in the Wikiversity article on [[Information is a public good: Designing experiments to improve government]].</ref> If this has an impact anywhere close to what is documented in the research that I cited above, it will substantially improve the prospects for broadly shared economic growth, while also reducing political polarization and violence and the prospects for war. Victor Picard, whom I interviewed December 13 of last year, recommends directing such subsidies to local multimedia centers managed perhaps by boards selected at random. These multimedia centers might help fund so-called documenters who observe public meetings and write notes that can be used by professional journalists in reports disseminated to a wider audience. Such multimedia centers might include journalism classes at local high schools and colleges, that may encourage migration to non-commercial social media, thereby also reducing teen suicides and political polarization and violence. === Other interviews === I feel a need to mention 5 other interviews. * August 19 of last year, I interviewed [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Francis Haugen]], who said that the shortest path to a click is anger or hate. Facebook executives had agreed that Facebook contributed to teen suicides and ethnic violence in several countries, including the [[w:Rohingya genocide|genocide of Rohingyan Muslims]] in Myanmar. Haugen was in charge of a department asked to reduce this problem. Then Facebook executives decided that if she were effective, it would reduce their profits. So they eliminated the department, and she became a whistleblower. * July 30 of last year, [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|Dean Baker, a co-founder of the Center for Economic and Policy Research, recommended changing Section 230 of Title 47 of the US Code]], which currently says that internet companies are not liable for content. Baker wants to change that so that internet companies are liable for content from which they make money boosting like print and broadcast media. Under the Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' (1964), but they would still be exempt when they are acting like common carriers, like a telephone company. * [[Evidence-informed public policy|Last July 31, Nick Hart, President and CEO of the Data Foundation]], discussed evidence-based public policy. He noted that President Trump in his first term signed bipartisan legislation requiring evidence-based public policy for decisions of the United States Congress. However, apparently, politicians are only allowed to consider evidence that has been broadly discussed by the major media. Otherwise, the major media can demonize them, like, what happened to the two senators who voted against the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]] in 1964. Research currently says that [[w:Immigration|immigrants benefit both the sending and the receiving countries]]. Some research on [[w:Sanctuary city|sanctuary cities]] find no difference between sanctuary and non-sanctuary cities. Other research insists that sanctuary cities actually have less crime and higher median incomes. The current [[w:Immigration policy of the second Trump administration|anti-immigrant campaign of the Trump administration]], in my judgment, is primarily the product of a conspiracy of silence by the major media, even the so-called liberal media, as well as well as the conservative media, prior to last November's election. There may have been some discussion of these points in some so-called liberal media outlets since last November, but those discussions have so far not been enough to allow that research to be seriously considered in Congress: The evidence is not sufficiently widely known to allow the Congress to actually consider it. * On August 28th I interviewed [[The role of the media in conflict|Doug Samuelson, who knows that before the first attack in a war, the different parties are polarized by their different media]]. He says he sometimes compares, for example, ''[[w:The New Republic|The New Republic]]'', known for its intellectual rigor and left-leaning political views, with the ''[[w:National Review|National Review]]'', an American conservative editorial magazine. Anything they agreed on was probably accurate. Disagreements clearly identified the spin. Primary drivers of any major conflict seem to be differences in the media that the different parties to conflict find credible. For example, supporters of Israel and supporters of Palestinians tend to find different media credible. Mira Sukharov, a Canadian Jew and professor of political science at Carleton University in Ottawa, Canada,<ref><!-- Mira Sukharov-->{{cite Q|Q136764001}}</ref> surveyed American Jews on Zionism: 58% self-reported as Zionists. 72% believed in a Jewish and democratic state. When asked if they believed in privileging Jews over non-Jews in Israel. 10% said yes, while 69% said no.<ref>Sucharov and Graves (2024).</ref> Few supporters of Israel, especially during the current war, have any awareness of the thousands of Palestinians, including hundreds of children, who have been routinely held for years without charges in Israeli prisons. Few supporters of Israel have any awareness of the [[w:Human rights violations against Palestinians by Israel|routine destruction or confiscation of Palestinian property by Israeli settlers protected by the Israeli military]]. Most of the media that supporters of Israel find credible rarely, if ever, report on such, but such is widely known among supporters of the Palestinians. Student protesters supporting the Palestinians see many reports of such in their social media feeds, which are largely suppressed by the major media in the United States. The major media coverage of such protests rarely mention the need to support the right of people peaceably to assemble, as supposedly secured by the First Amendment to the US Constitution. Supporters of Israel and supporters of Palestinians each have a long list of legitimate grievances against the other, but fail to understand how some of their actions have motivated the actions they deplore in their opposition. * Just over a month ago, [[Media Reform Coalition challenges anti-democratic media bias in the UK|I interviewed British journalist and researcher Dan Hind on the activities of the Media Reform Coalition]], which challenges anti-democratic bias in the UK. He claimed that before the US-led invasions of Afghanistan in 2001 and Iraq in 2003, elites in both the US and UK, the UK, told media executives that we were going to invade, and their job was to get the public behind those invasions. He also said that similar phenomena drove media coverage of the Cold War. I do not know if we can document such an elite conspiracy, but it is clear that major media organizations segment the media market in ways that increase political polarization and violence, and have contributed to the Cold War and the invasions of Afghanistan and Iraq and other questionable actions by the United States government at least since the end of the Korean conflict. === Acemoglu, Johnson, and Robinson === Finally, I feel a need to summarize, and to mention last year's Nobel Memorial Prize in Economics which went to [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]] for their leadership in documenting how the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to convince enough commoners that they could innovate and build a better world for themselves and others. In most other times and places in human history, religious authorities and others and, increasingly since the 1600s, media outlets have convinced the vast majority of humanity that they must accept their inferior law in life.<ref>Acemoglu and Robinson (2012).</ref> However, most economic growth, they note, has benefited only a few. To share the wealth more broadly, Acemoglu and Johnson recommend three things. 1. Change the narrative. 2. Build countervailing powers like organized labor. 3. Develop technical and policy solutions that benefit all.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> All three of these points can be helped with nonprofit media, like members of the Institute for Nonprofit News or community radio, because they are less likely to have conflicts of interest in reporting on anything that might offend people with power. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Acemoglu and Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Acemoglu and Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!-- Joshua Benton (9 April 2019). "When local newspapers shrink, fewer people bother to run for mayor". Nieman Foundation for Journalism -->{{cite Q|Q63127216}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2018) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2025) We have to talk-->{{cite Q|Q136126262}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)-->{{cite Q|Q131468166|authors=Richard R. John and Jonathan Silberstein-Loeb, eds.}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!--David A. Lieb (2024-04-04) " When voters say ‘no’ to new stadiums, what do professional sports teams do next?", AP-->{{cite Q|Q136763641}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}} * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Nobel Prize Committee (2002) The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 2002-->{{cite Q|Q136762831|author=Nobel Prize Committee}} * <!--Mira Sucharov and Spencer Graves (2024-06-04) "Mira Sucharov on Israel-Palestine"-->{{cite Q|Q136764029}} * <!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) -->{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Macroeconomics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> 9sisuizqzgt9j8gc5drtogin64lzas5 0 325723 2807005 2775740 2026-04-29T16:03:29Z DavidMCEddy 218607 link 2807005 wikitext text/x-wiki :''This discusses a 2025-11-20 interview with [[w:University of Denver|University of Denver]] journalism professor Kareem El Damanhoury<ref name=Daman><!--Kareem El Damanhoury-->{{cite Q|Q113752441}}</ref> about how different media cover similar issues. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2025-11-29 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Differences between media outlets including coverage of Gaza.webm|thumb|Differences between media outlets including coverage of Gaza per University of Denver Journalism Professor Kareem El Damanhoury]] [[File:Differences between media outlets including coverage of Gaza.ogg|thumb|29:00 mm:ss podcast from an interview conducted 2025-11-20 of University of Denver professor Kareem El Damanhoury about differences in how different media describe similar events including Gaza.]] University of Denver journalism professor Kareem El Damanhoury<ref name=Daman><!--Kareem El Damanhoury-->{{cite Q|Q113752441}}</ref> compares how the same or similar issues are framed differently in different media outlets. This includes especially comparing how [[w:Al Jazeera Media Network|Al Jazeera]], the [[w:BBC|BBC]]<ref>El Damanhoury et al. (2025).</ref> and [[w:Fox News|Fox]]<ref>El Damanhoury and Saleh (2024).</ref> have covered [[w:Gaza Strip|Gaza]]. El Damahoury is an Associate Professor of journalism at the University of Denver<ref name=Daman/> with numerous publications comparing different media outlets, e.g., on their coverage of Gaza and comparing the media in different counties in Colorado.<ref>El Damanhoury et al. (2022).</ref> He was born and raised in [[w:Egypt|Egypt]],<ref><!--Abdulaziz Al-Maosherji (undated) "Multilingual Faculty Spotlight: Dr. Kareem El Damanhoury-->{{cite Q|Q136829398}}</ref> earned a bachelors' from [[w:Cairo University|Cairo University]], an MA from [[w:Ohio University|Ohio University]] and a PhD from [[w:Georgia State University|Georgia State]].<ref><!--Kareem El Damanhoury, Georgia State alum-->{{cite Q|Q136830973}}</ref> He is an expert with the [[w:International Panel on the Information Environment|International Panel on the Information Environment]].<ref><!-- Kareem El Damanhoury, IPIE scientist-->{{cite Q|Q136831147}}</ref> Professor El Damanhoury is interviewed by Spencer Graves<ref name=Graves><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> and Doug Samuelson.<ref name = Samuelson><!--Douglas A. Samuelson-->{{cite Q|Q89781201}}</ref> == Highlights == Graves asked El Damanhoury to "describe the differences in the coverage of Gaza between Al Jazeera, the BBC and Fox." === Al Jazeera v. Fox === El Damanhoury began by summarizing his comparisons of media during the [[w:2014 Gaza War|2014 Gaza War]]. He said, "there was a big statistically significant difference when it comes to clarifying who died. ... Hamas is synonymous in the West to terrorism, and when you don't" differentiate between differentiate in reporting between militants, civilians, 2-year olds, etc., "this can have a huge implications." Similarly, in May 2021 there was another war. El Damanhoury compared how Al Jazeera and Fox used photographs. Both were highly dependent for images on [[w:Reuters|Reuters]], the [[w:Associated Press|AP (Associated Press)]], and [[w:Agence France-Presse|AFP (Agence France-Presse)]]. "Even though they were using the same images, the captions and the textual context was very different." For example, "You would see rockets in the air, and Al Jazeera would caption this as the resistance firing rockets at the occupation. While, on the other hand, you'll find Fox saying something along the lines of terrorists firing rockets at Tel Aviv." Both Al Jazeera and Fox adopted similar "David and Goliath" narratives but flipped the protagonist and the antagonist. Al Jazeera referred to people "dying in Gaza as martyrs. They would even draw comparisons to companions of Prophet Muhammad ... . [T]hey would call a mother of maybe like four kids who were killed by the Israeli military ... with the same name of the mother who was a companion to the Prophet over 1400 years ago. On the other hand, Fox would be hammering on the anti semitic tropes. ... as if the war that is happening in Gaza is actually impacting American Jews here physically and putting them in danger." In the current war, El Damanhoury is publishing multiple articles. A recent article discussed the sources being used. "It is one thing that you would be talking about a war in Gaza and only talking to, let's say, Republican senators and Israeli officials. It's another thing to be talking to people on the ground. While Al Jazeera was doing much more of talking to people on the ground who are impacted, and also leaning more into humanitarian organizations, BBC shied away from that, comparatively speaking". Graves asked, "Is it fair to summarize this by saying that collateral damage that our designated enemies commit proves to us that they are subhuman or at best criminally misled, while collateral damage that we commit is unfortunate but necessary"? El Damanhoury agreed. El Damanhoury mentioned an interview that President Trump gave to British media just few days ago in which he said that Ukrainian refugees "tend to assimilate well", implying that Palestinian refugees who may not be as white don't. === Local news === Graves requested a summary of El Damanhoury's research on local news. He said they've published reports on local media in both Colorado and New Jersey. "The Colorado trust wanted to pour in $5 million into the local news infrastructure in Colorado in order to bolster the news ecosystem, which, as you know, is struggling, not just in Colorado, but across the country. There are a lot of news deserts out there. ... [A news desert] is essentially a county that doesn't even have one newspaper ... covering its own city council, ... school districts, etc. [A lot of research] talks about how losing newspapers and counties can actually have devastating impacts on civic life, on politics, political engagement, ... the spread of misinformation and disinformation." El Damanhoury et al. got $100,000 from the Colorado Trust to study the local media in four counties focusing on the extent to which outlets actually described local events rather than something in another county. <ref>El Damanhoury et al. (2022).</ref> They focused on eight critical information needs described in a 2012 FCC report. These include health, education, environment and politics.<ref>Margolis (2012).</ref> They also considered [[w:Framing (social sciences)#Psychological roots of media framing research|episodic vs. thematic framing]]. For example, {{quote| [[w:George Floyd|George Floyd]] gets killed. A news outlet goes out there and talks about George Floyd and the incident that happens period. This is episodic. Why? Because it's talking about George Floyd as a separate incident, and it's just touching on it and not going to any other contextual information that is important in order to interpret what that happened. Another outlet goes out there and talks yes about the incident, yes. Talks about George Floyd. Yes, talks about what happened on that day or night. But then zooms out to ...[discuss the frequency of police killing Black people.] [W]hen you give the context, which is very important, this is thematic framing. ... We published an article in ''[[w:The Colorado Sun|The Colorado Sun]]'' [saying that very few articles touch] on critical information needs. For example, we came across radio outlets that were just copying and pasting Fox News articles. We came across newspapers that were essentially using newspapers from another big newspaper from the near city or nearest County. But the good news is that we found non traditional local news outlets that are doing fantastic work. One that jumps to mind is called the ''Southern Ute Drum''.<ref> Martinez (2019).</ref> This is a newspaper that is catering to the Native population in [[w:La Plata County, Colorado|La Plata County]]. They were doing a fantastic, stellar job at providing context, aka thematic framing, touching on things that are happening locally in their own community, doing original content by their own newspapers and touching on critical information needs. So in a nutshell, what we found is good news and bad news.}} === International Panel on the Information Environment === Graves asked about the [[w: International Panel on the Information Environment| International Panel on the Information Environment]] (IPIE). El Damanhoury said he did his [[w:Master's degree|master's]] at [[w:Ohio University|Ohio University]] in "communication and development", which is the use of communication media to spread social messages and achieve positive change. He studied entertainment education like ''[[w:Sesame Street|Sesame Street]]'': You get educated while also getting entertained. And he interned in 2014 at the [[w:United Nations|UN]] with their [[w:United Nations Department of Global Communications|office of public information]] and specifically with their [[w:United Nations Messengers of Peace|Messengers of Peace]] including [[w:Leonardo DiCaprio|Leonardo DiCaprio]]. IPIE focuses on how communication can help spread messages that sound the alarm but also give tangible information about how we can prevent a tragedy further down the line. === Humor === Samuelson asked, "What about what we laugh at? Late in the [[w:Presidency of George W. Bush|George W Bush administration]], somebody did a poll [asking for] the most trusted news source in the US, and the answer was [[w:Jon Stewart|Jon Stewart]] of ''[[w:The Daily Show|The Daily Show]]'', because they they got a good reputation for doing very careful fact checking before they made fun of anybody. ... Humor, I think conveys more than attempted straight reporting. What do you think?" El Damanhoury agreed but added that [[w:John Oliver|John Oliver]] {{quote|is offering investigative entertaining journalism. ... This is evidenced by how many times he has won [[w:Emmy Awards|Emmy Awards]] and [[w:Peabody Awards|Peabodies]]. ... The amount of fact checking and research that goes into any episode that he does is just mind boggling. There is that thing that is called [[w:Transportation theory (psychology)|narrative persuasion]]." For example, it's one thing to tell your listeners "Don't smoke". But it's more powerful if "I can show you somebody in a drama that you really associate with, and I can show you how they fell, and you can tear watching them, because there's something called [[w:Parasocial interaction|parasocial interaction]]", which is pseudo friendships. For example, "you'll be watching ''[[w:Rocky|Rocky]]'', and you'll be like, 'Come on, Rocky. Hit him.' You know that this guy is called Sylvester. His name is not Rocky, but you opted to calling him his nickname, because you now feel that you have a relationship with him." === Mispronouncing names === Samuelson noted that, "Just the name you call somebody makes a difference." After [[w:September 11 attacks|9-11-01]], "the name of the organization was being pronounced eight different ways in the meeting. I called up a friend who was fluent in Arabic. And said, 'How do you pronounce the name of that organization?' And he told me ... I noticed that some American officials were very deliberately" mispronouncing the name as a way of diminishing or dismissing them. There any significance to how that's pronounced? El Damanhoury connected this to the politician who just won the [[w:2025 New York City mayoral electionrace for Mayor of New York]]. ..."They kept mispronouncing his name, very deliberate ... to the extent that [[w:Elon Musk|Elon Musk]], I think, was tweeting at one point and even writing his name wrong. It's one thing to mispronounce the name and claim that you don't know how to pronounce it. It's another to actually spell it wrong." === News deserts === Graves asked about news deserts. El Damanhoury said, "That's the million dollar question." Graves replied, "I would say it's a several trillion dollar question." El Damanhoury agreed and added, "I was trying not to sound alarmist, I guess." El Damanhoury teaches a class at the University of Denver on "Media and terrorism", about how "terrorists" are covered across the spectrum, e.g., far fight, neo Nazis, environmentalists, etc. {{quote| Towards the end, we delve into how media can be used to counter those ideas that help lure youth into those rabbit holes. So one main thing is [[w:media literacy|media literacy]]. A Stanford study several years back found that middle schoolers and high schoolers were hardly able to differentiate between a paid ad and a news article. Another study found that folks were not able, or did not felt that they not have the chops in order to parse out what is true and what is not true. I am teaching a storytelling and reporting class in five prison facility correctional facilities in Colorado, and I was just grading before we hopped on a call, and one of my students was essentially saying, I try to avoid politics because I don't feel like I can discern what is factual and what is not. This is an epidemic ... that destroys the very premise of truth and facts and makes them very subjective. Media literacy is super important. Media Literacy ought not to be only provided in college. They ought to be, I argue, not only offered also in high school, but even in middle school.}} El Damanhoury and a colleague piloted a one week long media literacy class in high school to help students decide whether to pursue a career in media or engineering, business, physics, etc. {{quote| This is very important for the health of democracy and for the diffusing the extent of polarization that we've been living in for years now.}} Samuelson noted the need for reporters to consult two different sources. El Damanhoury agreed, noting that it's important for consumers as well as reporters to consult multiple sources -- especially with commercial media: If they are commercial, something sexy but not important will be covered. And it's important to know your sources. For example, the vast majority of his students did not know that [[w:RT (TV network)|RT]] was Russian. === In sum === Graves noted that we are about out of time and invited final comments. El Damanhoury replied, {{quote| There is a competition on our attention. That competition leads us to prioritize what we're consuming, Oftentimes we fall into the trap of [[w:cognitive dissonance|cognitive dissonance]]. We essentially try to gravitate to things that feed into our own predispositions. So I tell myself, and I suggest to anybody listening to get out of the comfort zone a bit. In my class of media and terrorism. I had once a former neo-Nazi come into class. He has since left and now has his own NGO working with youth to get them out of white supremacist and neo-Nazi groups ... . [S]tudents ... said, 'We've never had such an experience like this. When we talk to somebody who's been into that rabbit hole, came back and is now working to help people who are one where were in a position that he once was in.' ...[I]f I would have asked my students to go out there find the former neo-Nazi to speak to, probably nobody would have done that assignment. But ... getting that person into class was in a way, getting them out of their comfort zone. ... [T]the outcome was just stellar.}} == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Kareem El Damanhoury, David Coppini, Brittany Johnson, and Geneva Rodriguez (2022-06-06) "Local News in Colorado: Comparing Journalism Quality Across Four Counties"-->{{cite Q|Q136827833}} * <!--Kareem El Damanhoury and Faisal Saleh (2024-04-02) "Mediated Clash of Civilizations: Examining the Proximity-Visual Framing Nexus in Al Jazeera Arabic and Fox News’ Coverage of the 2021 Gaza War"-->{{cite Q|Q136827630}} * <!--Kareem El Damanhoury, Faisal Saleh, and Madeleine Lebovic (2025-01-16) "Covering the Israeli–Palestinian Conflict: A Critical Discourse Analysis of Al Jazeera English and BBC’s Online Reporting on the 2023 Gaza War"-->{{cite Q|Q136826072}} * <!--Daniel Margolis (2012) Review of the Literature Regarding Critical Information Needs of the American Public-->{{cite Q|Q136933744}} * <!--Fabian Martinez (2019-05-10) "A brief history of the Southern Ute Drum"-->{{cite Q|Q136934060}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Israel]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> anallj5avoqnk0j3v60s4z7nx43g42y 2807007 2807005 2026-04-29T16:10:07Z DavidMCEddy 218607 syntax 2807007 wikitext text/x-wiki :''This discusses a 2025-11-20 interview with [[w:University of Denver|University of Denver]] journalism professor Kareem El Damanhoury<ref name=Daman><!--Kareem El Damanhoury-->{{cite Q|Q113752441}}</ref> about how different media cover similar issues. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2025-11-29 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> [[File:Differences between media outlets including coverage of Gaza.webm|thumb|Differences between media outlets including coverage of Gaza per University of Denver Journalism Professor Kareem El Damanhoury]] [[File:Differences between media outlets including coverage of Gaza.ogg|thumb|29:00 mm:ss podcast from an interview conducted 2025-11-20 of University of Denver professor Kareem El Damanhoury about differences in how different media describe similar events including Gaza.]] [[w:University of Denver|University of Denver]] journalism professor Kareem El Damanhoury<ref name=Daman/> compares how the same or similar issues are framed differently in different media outlets. This includes especially comparing how [[w:Al Jazeera Media Network|Al Jazeera]], the [[w:BBC|BBC]]<ref>El Damanhoury et al. (2025).</ref> and [[w:Fox News|Fox]]<ref>El Damanhoury and Saleh (2024).</ref> have covered [[w:Gaza Strip|Gaza]]. El Damahoury is an Associate Professor of journalism at the University of Denver<ref name=Daman/> with numerous publications comparing different media outlets, e.g., on their coverage of Gaza and comparing the media in different counties in Colorado.<ref>El Damanhoury et al. (2022).</ref> He was born and raised in [[w:Egypt|Egypt]],<ref><!--Abdulaziz Al-Maosherji (undated) "Multilingual Faculty Spotlight: Dr. Kareem El Damanhoury-->{{cite Q|Q136829398}}</ref> earned a bachelors' from [[w:Cairo University|Cairo University]], an MA from [[w:Ohio University|Ohio University]] and a PhD from [[w:Georgia State University|Georgia State]].<ref><!--Kareem El Damanhoury, Georgia State alum-->{{cite Q|Q136830973}}</ref> He is an expert with the [[w:International Panel on the Information Environment|International Panel on the Information Environment]].<ref><!-- Kareem El Damanhoury, IPIE scientist-->{{cite Q|Q136831147}}</ref> Professor El Damanhoury is interviewed by Spencer Graves<ref name=Graves><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> and Doug Samuelson.<ref name = Samuelson><!--Douglas A. Samuelson-->{{cite Q|Q89781201}}</ref> == Highlights == Graves asked El Damanhoury to "describe the differences in the coverage of Gaza between Al Jazeera, the BBC and Fox." === Al Jazeera v. Fox === El Damanhoury began by summarizing his comparisons of media during the [[w:2014 Gaza War|2014 Gaza War]]. He said, "there was a big statistically significant difference when it comes to clarifying who died. ... Hamas is synonymous in the West to terrorism, and when you don't" differentiate between differentiate in reporting between militants, civilians, 2-year olds, etc., "this can have a huge implications." Similarly, in May 2021 there was another war. El Damanhoury compared how Al Jazeera and Fox used photographs. Both were highly dependent for images on [[w:Reuters|Reuters]], the [[w:Associated Press|AP (Associated Press)]], and [[w:Agence France-Presse|AFP (Agence France-Presse)]]. "Even though they were using the same images, the captions and the textual context was very different." For example, "You would see rockets in the air, and Al Jazeera would caption this as the resistance firing rockets at the occupation. While, on the other hand, you'll find Fox saying something along the lines of terrorists firing rockets at Tel Aviv." Both Al Jazeera and Fox adopted similar "David and Goliath" narratives but flipped the protagonist and the antagonist. Al Jazeera referred to people "dying in Gaza as martyrs. They would even draw comparisons to companions of Prophet Muhammad ... . [T]hey would call a mother of maybe like four kids who were killed by the Israeli military ... with the same name of the mother who was a companion to the Prophet over 1400 years ago. On the other hand, Fox would be hammering on the anti semitic tropes. ... as if the war that is happening in Gaza is actually impacting American Jews here physically and putting them in danger." In the current war, El Damanhoury is publishing multiple articles. A recent article discussed the sources being used. "It is one thing that you would be talking about a war in Gaza and only talking to, let's say, Republican senators and Israeli officials. It's another thing to be talking to people on the ground. While Al Jazeera was doing much more of talking to people on the ground who are impacted, and also leaning more into humanitarian organizations, BBC shied away from that, comparatively speaking". Graves asked, "Is it fair to summarize this by saying that collateral damage that our designated enemies commit proves to us that they are subhuman or at best criminally misled, while collateral damage that we commit is unfortunate but necessary"? El Damanhoury agreed. El Damanhoury mentioned an interview that President Trump gave to British media just few days ago in which he said that Ukrainian refugees "tend to assimilate well", implying that Palestinian refugees who may not be as white don't. === Local news === Graves requested a summary of El Damanhoury's research on local news. He said they've published reports on local media in both Colorado and New Jersey. "The Colorado trust wanted to pour in $5 million into the local news infrastructure in Colorado in order to bolster the news ecosystem, which, as you know, is struggling, not just in Colorado, but across the country. There are a lot of news deserts out there. ... [A news desert] is essentially a county that doesn't even have one newspaper ... covering its own city council, ... school districts, etc. [A lot of research] talks about how losing newspapers and counties can actually have devastating impacts on civic life, on politics, political engagement, ... the spread of misinformation and disinformation." El Damanhoury et al. got $100,000 from the Colorado Trust to study the local media in four counties focusing on the extent to which outlets actually described local events rather than something in another county. <ref>El Damanhoury et al. (2022).</ref> They focused on eight critical information needs described in a 2012 FCC report. These include health, education, environment and politics.<ref>Margolis (2012).</ref> They also considered [[w:Framing (social sciences)#Psychological roots of media framing research|episodic vs. thematic framing]]. For example, {{quote| [[w:George Floyd|George Floyd]] gets killed. A news outlet goes out there and talks about George Floyd and the incident that happens period. This is episodic. Why? Because it's talking about George Floyd as a separate incident, and it's just touching on it and not going to any other contextual information that is important in order to interpret what that happened. Another outlet goes out there and talks yes about the incident, yes. Talks about George Floyd. Yes, talks about what happened on that day or night. But then zooms out to ...[discuss the frequency of police killing Black people.] [W]hen you give the context, which is very important, this is thematic framing. ... We published an article in ''[[w:The Colorado Sun|The Colorado Sun]]'' [saying that very few articles touch] on critical information needs. For example, we came across radio outlets that were just copying and pasting Fox News articles. We came across newspapers that were essentially using newspapers from another big newspaper from the near city or nearest County. But the good news is that we found non traditional local news outlets that are doing fantastic work. One that jumps to mind is called the ''Southern Ute Drum''.<ref> Martinez (2019).</ref> This is a newspaper that is catering to the Native population in [[w:La Plata County, Colorado|La Plata County]]. They were doing a fantastic, stellar job at providing context, aka thematic framing, touching on things that are happening locally in their own community, doing original content by their own newspapers and touching on critical information needs. So in a nutshell, what we found is good news and bad news.}} === International Panel on the Information Environment === Graves asked about the [[w: International Panel on the Information Environment| International Panel on the Information Environment]] (IPIE). El Damanhoury said he did his [[w:Master's degree|master's]] at [[w:Ohio University|Ohio University]] in "communication and development", which is the use of communication media to spread social messages and achieve positive change. He studied entertainment education like ''[[w:Sesame Street|Sesame Street]]'': You get educated while also getting entertained. And he interned in 2014 at the [[w:United Nations|UN]] with their [[w:United Nations Department of Global Communications|office of public information]] and specifically with their [[w:United Nations Messengers of Peace|Messengers of Peace]] including [[w:Leonardo DiCaprio|Leonardo DiCaprio]]. IPIE focuses on how communication can help spread messages that sound the alarm but also give tangible information about how we can prevent a tragedy further down the line. === Humor === Samuelson asked, "What about what we laugh at? Late in the [[w:Presidency of George W. Bush|George W Bush administration]], somebody did a poll [asking for] the most trusted news source in the US, and the answer was [[w:Jon Stewart|Jon Stewart]] of ''[[w:The Daily Show|The Daily Show]]'', because they they got a good reputation for doing very careful fact checking before they made fun of anybody. ... Humor, I think conveys more than attempted straight reporting. What do you think?" El Damanhoury agreed but added that [[w:John Oliver|John Oliver]] {{quote|is offering investigative entertaining journalism. ... This is evidenced by how many times he has won [[w:Emmy Awards|Emmy Awards]] and [[w:Peabody Awards|Peabodies]]. ... The amount of fact checking and research that goes into any episode that he does is just mind boggling. There is that thing that is called [[w:Transportation theory (psychology)|narrative persuasion]]." For example, it's one thing to tell your listeners "Don't smoke". But it's more powerful if "I can show you somebody in a drama that you really associate with, and I can show you how they fell, and you can tear watching them, because there's something called [[w:Parasocial interaction|parasocial interaction]]", which is pseudo friendships. For example, "you'll be watching ''[[w:Rocky|Rocky]]'', and you'll be like, 'Come on, Rocky. Hit him.' You know that this guy is called Sylvester. His name is not Rocky, but you opted to calling him his nickname, because you now feel that you have a relationship with him." === Mispronouncing names === Samuelson noted that, "Just the name you call somebody makes a difference." After [[w:September 11 attacks|9-11-01]], "the name of the organization was being pronounced eight different ways in the meeting. I called up a friend who was fluent in Arabic. And said, 'How do you pronounce the name of that organization?' And he told me ... I noticed that some American officials were very deliberately" mispronouncing the name as a way of diminishing or dismissing them. There any significance to how that's pronounced? El Damanhoury connected this to the politician who just won the [[w:2025 New York City mayoral electionrace for Mayor of New York]]. ..."They kept mispronouncing his name, very deliberate ... to the extent that [[w:Elon Musk|Elon Musk]], I think, was tweeting at one point and even writing his name wrong. It's one thing to mispronounce the name and claim that you don't know how to pronounce it. It's another to actually spell it wrong." === News deserts === Graves asked about news deserts. El Damanhoury said, "That's the million dollar question." Graves replied, "I would say it's a several trillion dollar question." El Damanhoury agreed and added, "I was trying not to sound alarmist, I guess." El Damanhoury teaches a class at the University of Denver on "Media and terrorism", about how "terrorists" are covered across the spectrum, e.g., far fight, neo Nazis, environmentalists, etc. {{quote| Towards the end, we delve into how media can be used to counter those ideas that help lure youth into those rabbit holes. So one main thing is [[w:media literacy|media literacy]]. A Stanford study several years back found that middle schoolers and high schoolers were hardly able to differentiate between a paid ad and a news article. Another study found that folks were not able, or did not felt that they not have the chops in order to parse out what is true and what is not true. I am teaching a storytelling and reporting class in five prison facility correctional facilities in Colorado, and I was just grading before we hopped on a call, and one of my students was essentially saying, I try to avoid politics because I don't feel like I can discern what is factual and what is not. This is an epidemic ... that destroys the very premise of truth and facts and makes them very subjective. Media literacy is super important. Media Literacy ought not to be only provided in college. They ought to be, I argue, not only offered also in high school, but even in middle school.}} El Damanhoury and a colleague piloted a one week long media literacy class in high school to help students decide whether to pursue a career in media or engineering, business, physics, etc. {{quote| This is very important for the health of democracy and for the diffusing the extent of polarization that we've been living in for years now.}} Samuelson noted the need for reporters to consult two different sources. El Damanhoury agreed, noting that it's important for consumers as well as reporters to consult multiple sources -- especially with commercial media: If they are commercial, something sexy but not important will be covered. And it's important to know your sources. For example, the vast majority of his students did not know that [[w:RT (TV network)|RT]] was Russian. === In sum === Graves noted that we are about out of time and invited final comments. El Damanhoury replied, {{quote| There is a competition on our attention. That competition leads us to prioritize what we're consuming, Oftentimes we fall into the trap of [[w:cognitive dissonance|cognitive dissonance]]. We essentially try to gravitate to things that feed into our own predispositions. So I tell myself, and I suggest to anybody listening to get out of the comfort zone a bit. In my class of media and terrorism. I had once a former neo-Nazi come into class. He has since left and now has his own NGO working with youth to get them out of white supremacist and neo-Nazi groups ... . [S]tudents ... said, 'We've never had such an experience like this. When we talk to somebody who's been into that rabbit hole, came back and is now working to help people who are one where were in a position that he once was in.' ...[I]f I would have asked my students to go out there find the former neo-Nazi to speak to, probably nobody would have done that assignment. But ... getting that person into class was in a way, getting them out of their comfort zone. ... [T]the outcome was just stellar.}} == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Kareem El Damanhoury, David Coppini, Brittany Johnson, and Geneva Rodriguez (2022-06-06) "Local News in Colorado: Comparing Journalism Quality Across Four Counties"-->{{cite Q|Q136827833}} * <!--Kareem El Damanhoury and Faisal Saleh (2024-04-02) "Mediated Clash of Civilizations: Examining the Proximity-Visual Framing Nexus in Al Jazeera Arabic and Fox News’ Coverage of the 2021 Gaza War"-->{{cite Q|Q136827630}} * <!--Kareem El Damanhoury, Faisal Saleh, and Madeleine Lebovic (2025-01-16) "Covering the Israeli–Palestinian Conflict: A Critical Discourse Analysis of Al Jazeera English and BBC’s Online Reporting on the 2023 Gaza War"-->{{cite Q|Q136826072}} * <!--Daniel Margolis (2012) Review of the Literature Regarding Critical Information Needs of the American Public-->{{cite Q|Q136933744}} * <!--Fabian Martinez (2019-05-10) "A brief history of the Southern Ute Drum"-->{{cite Q|Q136934060}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Israel]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> q4xx0ygppyb6lwi9rd6al9ds8m19kx3 2 326765 2807041 2806941 2026-04-29T18:04:59Z Dc.samizdat 2856930 /* Hypercubes */ 2807041 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells it has two sets of 6 pairwise orthogonal square central planes. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} cjj6otcxsfwg3qrld7jrku4kj7alr82 2807042 2807041 2026-04-29T18:54:16Z Dc.samizdat 2856930 2807042 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90 apart in one angle, and either 0 or 90 apart in the other angle. They are 90 apart in both angles if and only if they are completely orthogonal; otherwise they are 0 apart in one angle, they intersect in an axis, and they lie in the same 3-dimensional hyperplane. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 2ogu06ey3sukz3svl8gyyo6aujlmu3e 2807044 2807042 2026-04-29T19:42:05Z Dc.samizdat 2856930 2807044 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90 degrees apart in one angle, and either 0 degrees or 90 degrees apart in the other angle. They are 90 degrees apart in both angles if and only if they are completely orthogonal, 90 degrees apart by isoclinic rotation, with no vertices in common. Otherwise they are 0 degrees apart in one angle, 90 degrees apart by simple rotation, and they intersect in an axis and lie in a common 3-dimensional hyperplane. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 5914c4s19th4d84rpuadayrosp0i61k 2807045 2807044 2026-04-29T19:57:14Z Dc.samizdat 2856930 2807045 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90 degrees apart in one angle, and either 0 degrees or 90 degrees apart in the other angle. They are 90 degrees apart in both angles if and only if they are completely orthogonal, 90 degrees apart by isoclinic rotation, with no vertices in common. Otherwise they are 0 degrees apart in one angle, 90 degrees apart by simple rotation, and they intersect in an axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60 degrees apart by isoclinic rotation. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} d6lm9uyd4vdeenj0rywnnbcl5y4rbto 2807047 2807045 2026-04-29T20:00:59Z Dc.samizdat 2856930 2807047 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one angle, 90° apart by simple rotation, and they intersect in an axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 7nh724rs6btt24z9079g4isql7yfsi9 2807048 2807047 2026-04-29T20:03:00Z Dc.samizdat 2856930 2807048 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one angle, 90° apart by simple rotation, and they intersect in an axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} j8esr4nxy6rctun0xlrpca3o08yckk5 2807049 2807048 2026-04-29T20:19:05Z Dc.samizdat 2856930 2807049 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart. Not only their square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ajb0gv14yeev3rsr1z7imamve1h3sqd 2807050 2807049 2026-04-29T20:23:51Z Dc.samizdat 2856930 /* Hypercubes */ 2807050 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes in each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart. Not only their square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} dfqwset3ji5sl10iqgfwuplpz6a93l4 2807051 2807050 2026-04-29T20:25:40Z Dc.samizdat 2856930 2807051 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart. Not only their square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} r72m4zst797fzrme3cf8pzy22fdauic 2807053 2807051 2026-04-29T20:33:12Z Dc.samizdat 2856930 /* Hypercubes */ 2807053 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane (they belong to the same cube). Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart. Not only their square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 8t4696f4i5s030cdnzgeohncfkeeo9s 2807054 2807053 2026-04-29T20:36:24Z Dc.samizdat 2856930 2807054 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane (they belong to the same cube). Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} eks00bdznpifu8g3dtjfo5eep8shg4q 2807055 2807054 2026-04-29T20:39:22Z Dc.samizdat 2856930 /* Hypercubes */ 2807055 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane (they belong to the same cube). Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their pairs of square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 1bm6skkkdaotopxdfxw0wupqatax1pe 2807056 2807055 2026-04-29T20:42:54Z Dc.samizdat 2856930 /* Hypercubes */ 2807056 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their pairs of square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} pyxlbaqit51mieimc0t2b9irn4q6tpp 2807057 2807056 2026-04-29T20:46:37Z Dc.samizdat 2856930 2807057 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} rn2ynsleyfwz9u0uymy9cflprd9d2b1 2807058 2807057 2026-04-29T21:00:08Z Dc.samizdat 2856930 2807058 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 2p3h4owbiaig5eqbfcrtcqajkbrrtvo 2807060 2807058 2026-04-29T21:24:26Z Dc.samizdat 2856930 2807060 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common, and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane. Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation,{{Efn|A simple rotation is a double rotation in which one of the two angles of rotation is 0°, so one of the completely orthogonal invariant planes of rotation does not move. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 1ebtqoek931ifqhy24hhv7hn6e027fq 2807061 2807060 2026-04-29T22:42:51Z Dc.samizdat 2856930 2807061 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} ff1ck5ufqaqs6mniw6h5xn85xqu8wvc 2807062 2807061 2026-04-29T23:19:24Z Dc.samizdat 2856930 2807062 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 54sluyeywxp90ag05kg7xvmcugfi0zi 2807063 2807062 2026-04-29T23:25:03Z Dc.samizdat 2856930 /* Hypercubes */ 2807063 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} gwn2n76mioivh68a60y7yfqcf44cjgo 2807064 2807063 2026-04-29T23:35:58Z Dc.samizdat 2856930 /* Hypercubes */ 2807064 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once, and returns to its original position, but it does not visit the vertex positions of the other 16-cell. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 0rys1wxuqgtc9t180rhoex1kl0e4m6o 2807065 2807064 2026-04-29T23:53:52Z Dc.samizdat 2856930 /* Hypercubes */ 2807065 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once, and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The geodesic orbits of the vertices are disjoint circular helixes, and the circular helixes of corresponding vertices in the two 16-cells are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} s0hn8iwxo4rpuxrlgq9b0vh4rs0z5vf 2807066 2807065 2026-04-30T00:38:22Z Dc.samizdat 2856930 2807066 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once, and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The geodesic orbits of the vertices are disjoint circular helixes, and the circular helixes of corresponding vertices in the two 16-cells are Clifford parallel objects. We can also rotate the tesseract isoclinically in a new way, by 60° in two completely orthogonal invariant hexagonal central planes, with a different effect on the alternate 16-cells. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} s78uv9gwzyfqgbs0zkwxn6ovq14fp1r 2807085 2807066 2026-04-30T05:41:31Z Dc.samizdat 2856930 /* Hypercubes */ 2807085 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - April 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two smaller chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by <math>5^2</math> non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times <math>5^2</math> (75) disjoint instances of itself in the 600-point (120-cell), which contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point (24-cell), and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Notice that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, so we obtain a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, so we obtain a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <small><math>1/\sqrt{2}</math></small>. The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell has 3 such Petrie octagons, which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <small><math>\sqrt{2}</math></small> edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length: :<math>r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552</math> none of which chords except <math>r_1=\sqrt{2}</math> occur in the 16-cell. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <small><math>n</math></small> is <small><math>\sqrt{n}</math></small>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The cuboctahedron and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length <small><math>\sqrt{2}</math></small> that we needed to build the unit-radius 16-cell. For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the <small><math>\sqrt{2}</math></small> edges of two unit-radius 16-cells, which are also the edges of the square central planes. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship of two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one angle, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}} Pairs of square central planes in alternate 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. They are not orthogonal or parallel so they intersect in a line somewhere, but they have no vertices in common. They cannot reach each other by simple rotation, and they have no 3-dimensional hyperplane in common. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only their corresponding square central planes but the alternate 16-cells themselves are Clifford parallel objects. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once, and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The geodesic orbits of the vertices are disjoint circular helixes, and the circular helixes of corresponding vertices in the two 16-cells are Clifford parallel objects. == The 24-cell == ... == The 600-cell == ... == Finally, the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} s0hn8iwxo4rpuxrlgq9b0vh4rs0z5vf 0 326829 2807009 2787839 2026-04-29T16:11:01Z Lar35896 3044450 name 2807009 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Sarah Jordan Lesson Plan Draft One]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Mitchell McDaniel Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|Raymond Nieves Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|Sahil Bashir Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Elián Fabbrini Lesson Plan]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] 8k8dyp2nxji2mv89iuxknvtmdwi3l2j 2807010 2807009 2026-04-29T16:11:57Z RayN53 3044043 Changed my title 2807010 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Sarah Jordan Lesson Plan Draft One]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Mitchell McDaniel Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|Sahil Bashir Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Elián Fabbrini Lesson Plan]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] g300zy8fcpwg9ikzuovgf7z6c45rf01 2807011 2807010 2026-04-29T16:12:09Z Lar35896 3044450 name 2807011 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Sarah Jordan Lesson Plan Draft One]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Mitchell McDaniel Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Elián Fabbrini Lesson Plan]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] i933u7kj98w9c10iuk12rev3b9accdr 2807012 2807011 2026-04-29T16:12:42Z UrAvgEMATer 3044443 2807012 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Sarah Jordan Lesson Plan Draft One]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Mitchell McDaniel Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] qfbq27lag8o3z88mug14sz8qi8ddyzs 2807013 2807012 2026-04-29T16:12:53Z ~2026-26080-89 3069312 2807013 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Mitchell McDaniel Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] a5nkcpzb5422as27r30jbleypptjmm7 2807014 2807013 2026-04-29T16:14:31Z Mbm41048 3043729 2807014 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration and Comparison of Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] d06af5nypduq2n783nuyk803g6tp8v9 2807016 2807014 2026-04-29T16:15:00Z ~2026-26080-89 3069312 2807016 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration and Comparison of Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] 6lcttk5uk45trz5ol4jnc250ufze0ne 2807017 2807016 2026-04-29T16:15:11Z NDG 3006541 Reverted edits by [[Special:Contribs/~2026-26080-89|~2026-26080-89]] ([[User talk:~2026-26080-89|talk]]) to last version by Mbm41048: test edits, please use the sandbox 2807014 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration and Comparison of Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Ella Barrett Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] d06af5nypduq2n783nuyk803g6tp8v9 2807018 2807017 2026-04-29T16:15:43Z ~2026-70382-1 3048442 2807018 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration and Comparison of Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Lilli Engledow Task Plan]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] c8wfhh6eah93s0f8r5bjg5svoky847y 2807019 2807018 2026-04-29T16:16:07Z ~2026-25110-59 3068253 2807019 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration and Comparison of Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Sydney Stampfli- Task Plan]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Comparing Linear Functions Task]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] m86kueyxx5zlmpe41kb8kdue8hxa1hu 2807020 2807019 2026-04-29T16:16:13Z ~2026-26059-54 3069313 2807020 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration and Comparison of Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Creating exponential equations in one variable and using them to solve problems]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Comparing Linear Functions Task]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] mgtg5vmt1df2h32qm3k1hijjw9kmrt1 2807021 2807020 2026-04-29T16:16:45Z Mbm41048 3043729 /* Examples */ 2807021 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Ashley Abraham Task Plan]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration of Linear Functions in Context Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Creating exponential equations in one variable and using them to solve problems]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Comparing Linear Functions Task]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] ejppuezzyzl9chqhbzuii243o7e6y1e 2807023 2807021 2026-04-29T16:17:22Z ~2026-26130-81 3069314 /* Examples */ 2807023 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Creating A Linear Function]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration of Linear Functions in Context Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Creating exponential equations in one variable and using them to solve problems]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Jenna Weinberger Task Plan]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Comparing Linear Functions Task]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] kd8pcmmd8imbza3lglbpl7dihaz2sww 2807024 2807023 2026-04-29T16:17:34Z Jenna Weinberger 3044068 2807024 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Creating A Linear Function]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration of Linear Functions in Context Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Creating exponential equations in one variable and using them to solve problems]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Ethan Kruger Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Interpreting Quadratic Functions with Projectile Motion]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Comparing Linear Functions Task]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] f0yg1z7hsytm4t27i4fi559b8gtz6vq 2807025 2807024 2026-04-29T16:19:40Z Esk297 3044098 Changed Ethan's Title 2807025 wikitext text/x-wiki Prospective teachers often are asked to create or modify lesson plans as part of their training. == Examples == ''Please make sure that you create your page as a subpage of "Lesson plans created by prospective teachers", like so: "Lesson plans created by prospective teachers/[Plan Name]".'' *[[Lesson plans created by prospective teachers/Ashley Abraham Task Plan|Creating A Linear Function]] *[[Lesson plans created by prospective teachers/Sarah Jordan Task Plan|Different ways of thinking of a Quadratic Task: Steel Cables]] [https://outlookuga-my.sharepoint.com/:w:/g/personal/snj18898_uga_edu/IQDosd1jPwksTLzFmcE3pjBnAWXjHhHfil6_95Rw4FNgirk?e=Bc9VAv] *[[Lesson plans created by prospective teachers/Mitchell McDaniel Task Plan|Exploration of Linear Functions in Context Task Plan]] *[[Lesson plans created by prospective teachers/Sydney Stampfli|Creating exponential equations in one variable and using them to solve problems]] *[[Lesson plans created by prospective teachers/Raymond Nieves Task Plan|System of Equations Task Plan]] [https://drive.google.com/file/d/13pIvZUJwXYcefUC580ddrnwoaqvFrI0Q/view?usp=sharing] * [[Lesson plans created by prospective teachers/Ethan Kruger Task Plan|Quadratic Expressions and the Handshake Problem: A Task Plan]] * [[Lesson plans created by prospective teachers/Jenna Weinberger Task Plan|Interpreting Quadratic Functions with Projectile Motion]] *[[Lesson plans created by prospective teachers/Sahil Bashir Task Plan|One-Step Equations From Context Task Plan]] *[[Lesson plans created by prospective teachers/Lilli Engledow Task Plan|Comparing Linear Functions Task]] *[[Lesson plans created by prospective teachers/Elián Fabbrini Task Plan|Solving Systems of Equations by Algebra, Table, and Graph]] *[[Lesson plans created by prospective teachers/Layne Rouse Task Plan|Graphing and Interpreting Linear Functions Task Plan]] *[[Lesson plans created by prospective teachers/Ella Barrett Task Plan|Counting Stairs Task Plan]] [[Category:Lesson plans created by prospective teachers| ]] al5jsjwr54yqxs0uroifw03j7yroody 0 326955 2807046 2806917 2026-04-29T19:59:00Z Scogdill 1331941 2807046 wikitext text/x-wiki [[File:Kaiserin Elisabeth im Morgenlicht.jpg|alt=Old painting of a beautiful white woman with her very long hair tied at below her neck like a scarf|thumb|Winterhalter, "Empress Elisabeth in the Morning Light," 1864]] ==Overview== Empress Elisabeth was politically progressive, especially compared to her husband, Franz Josef I, and his mother (and her aunt) Archduchess Sophia, who were committed to absolute and hereditary monarchy and were never able to see the potential value of a compromise or the development of a constitutional monarchy. Elisabeth had been raised in an unstructured and permissive environment with a lot of personal freedom. She was more alienated from the very conservative court than she was from the people in the Austrian and Hungarian empire who were demanding a constitutional monarchy. At the beginning of her married life, Elisabeth lived in at the Habsburg court in Vienna in what was for her a toxic environment that undermined her health and agency. She was surprised to find herself pregnant and developed a fear of going down steep stairs.<ref name=":4" /> She had a persistent cough and, after her daughter Sophie died at 2 in 1857, was unable to eat, to the point that sometimes she was too weak and her feet and legs were so swollen she could not walk without help. In November 1860, when Elisabeth was very ill, Queen Victoria sent her yacht to take her from Antwerp to Madeira, where she did in fact get better. Franz Josef I was a Hapsburg but not the Holy Roman Emperor: the Holy Roman Empire was "formally brought ... to an end" by Napoleon in 1806.<ref name=":9">Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books: https://books.apple.com/us/book/victoria/id828766078.</ref>{{rp|68 of 1204}} He, King Edward VII, and Emperor William II of Germany were the most powerful monarchs in Europe, but Kaiser William was "showy, vainglorious, unstable" and Franz Josef was "imposssibly remote."<ref name=":8">Van der Kiste, John. ''Queen Victoria's Children''. The History Press, 2011 (1986).</ref>{{rp|247 of 300}} According to A. N. Wilson, Prince "Albert wanted to be friends with the Emperor of Austria, perhaps one day to see him as the figurehead of a united Germany."<ref name=":9" />{{rp|246 of 1204}} After the death of her first child, Elisabeth's next two children were also taken away from her and raised by the Archduchess Sophia, so their relationship with their mother was never close. Elisabeth herself raised her last child, Valerie, born in 1868 in Buda, Hungary.<ref name=":0">{{Cite journal|date=2026-01-13|title=Archduchess Marie Valerie of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduchess_Marie_Valerie_of_Austria&oldid=1332726352|journal=Wikipedia|language=en}}</ref> Although he was politically reactionary, not faithful and unable to withstand his mother's influence, Franz Joseph actually seems to have loved Elisabeth, who found him incompatible. Their relationship improved over the course of their lives, and their letters suggest it had developed into a genuine friendship when Elisabeth was assassinated in 1898.<ref name=":4" /> She read widely and collected the letters of Heinrich Heine.<ref name=":4" /> She wrote personal poetry, which she published, that was inspired by Heine. She wrote a "humorous" poem about Albert Edward, Prince of Wales called "There is somebody coming upstairs."<ref name=":10">Káli-Rozmis, Barbara. "The Visits of Empress Elisabeth of Austria in Britain and its Impact on the English and Irish People." Freeside Europe Academic Journal 2020, 1 [https://www.freesideeurope.com/articles/the-visits-of-empress-elisabeth-of-austria-in-britain-and-its-impact-on-the-english-and-irish-people-2 DOI 10.51313/alumni-2020-2]. https://www.freesideeurope.com/images/pdf/pdf_2.pdf</ref>{{rp|4}} She also traveled widely, spending much of her time away from Vienna once her first 3 children were born. She was not constrained by what Queen Victoria would have considered her "duty" as monarch, spent a great deal of time away from court and did not take part in state occasions, which she found boring. Her English was very good. Elisabeth's affinity for Hungary made her a successful advocate and negotiator between it and Austria. She may have suffered from depression and possibly disordered eating — anorexia, compulsive exercising and bulimia can be inferred. Later in her life, she took up smoking.[[File:Winterhalter Elisabeth.jpg|alt=Old painting of a wealthy queen wearing a fluffy pink dress and lots of jewelry|thumb|Empress Elisabeth by Winterhalter, 1865, in a dress by Maison Worth]] As an adult Elisabeth was a client of Charles Frederick Worth of Paris.<ref>{{Cite journal|title=Charles Frederick Worth|url=https://en.wikipedia.org/w/index.php?title=Charles_Frederick_Worth&oldid=1322100685|journal=Wikipedia|date=2025-11-14|language=en}}</ref> The 1865 Franz Xaver Winterhalter portrait of her (top right) shows her in a Worth pink tulle ball gown with her signature diamond stars in her hair. She practiced tight lacing, getting her waist down to 16 inches in 1859–1860. One lady in waiting said in 1863 she was no longer tight lacing (perhaps she had stopped using French leather corsets), but her waist was 18¹/₂–19¹/₂ inches for the rest of her life.<ref name=":4" /> Her youngest daughter was afraid of Queen Victoria when she met her because of the size of the older woman's body and, presumably, her mother's obsesssions. The leather corsets, which are known to have been used by Parisian sex workers, may have been part of a "bordello-culture" interest of Sisi's,<ref name=":11">{{Cite web|url=https://www.andreanolen.com/home/empress-sisi-and-prostitution|title=Empress 'Sisi' and Prostitution|date=2021-08-17|website=Andrea Nolen|language=en-US|access-date=2026-03-27}}</ref> but the tight lacing is also consistent with disordered eating, in this case actively attempting to be extremely thin. Elisabeth dressed opulently, although as a very skilled horsewoman, she often wore riding habits, and was very fashion forward, owning a cage early and giving them up for the more fitted look as soon as Worth and fashion did.<ref name=":4" /> She was "possibly the best-known [horsewoman] of her time<ref name=":4" /> and, like Alexandra, Princess of Wales, wore closely fitting riding habits as normal day wear. Her clothing fit her so perfectly that it was said that she was sewn into it,<ref name=":10" />{{rp|4}} and certainly she had the staff for such a practice. Her riding habit was dark blue with gold buttons, accessorized with a blue veil and an orange fan.<ref name=":10" />{{rp|4}} Elisabeth modeled her stylish look on Eugénie:<blockquote>Her spectacular crinoline evening gowns, modeled on Eugénie’s, were so bouffant that on one occasion the archbishop of Milan became entangled with her dress merely by passing her on a staircase, and to her barely repressed giggles, could only be liberated by tearing off some of the outer folds.<ref name=":6">Goldstone, Nancy. ''The Rebel Empresses: Elisabeth of Austria and Eugénie of France, Power and Glamour in the Struggle for Europe''. Little, Brown, 2025.</ref>{{rp|322 of 909}}</blockquote> Sisi collected photographs of beautiful women from all classes, including sex workers, in "albums of beauty." Christina Catherine Martinez says she had 2,000 photographs in her collection.<ref name=":11" /> Elisabeth's relationship with Victoria was not very warm or close, although Elisabeth spent a lot of time in the UK and Ireland. Perhaps her many trips to Ireland and few visits to Victoria had an effect on the older woman. ==Also Known As== * Family name ** Hers: House of Wittelsbach, Bavaria ** His: Habsburg * Elisabeth Amalie Eugenie * Sisi * Elisabeth of Bavaria * For incognito travel: Countess Hohenembs<ref name=":4" /> ==Acquaintances, Friends and Enemies== ===Elisabeth=== ==== Acquaintances ==== * [[Social Victorians/People/Mary Todd Lincoln|Mary Todd Lincoln]] * [[Social Victorians/People/Queen Victoria|Queen Victoria]] * Vicky? ==== Friends ==== * Count Gyula Andrássy * Albert Edward, Prince of Wales<ref name=":10" /> (2) * Alexandra, Princess of Wales * Lord Spencer, former Lord Lieutenant of Ireland<ref name=":10" /> (3) * Captain William George (Bay) Middleton, Scots horseman and equerry to Spencer<ref name=":10" /> (3) ==== Enemies ==== * Princess Pauline von Metternich, close friend of [[Social Victorians/People/Eugenie of France|Eugénie of France]] ==== Members of Her Retinue and Employees ==== * Franziska (Fanny) Feifalik, hairdresser, "enge Vertraute [close confidant]" and "ihr Double [body double]" (1863–1896)<ref>{{Cite journal|date=2025-08-19|title=Franziska Feifalik|url=https://de.wikipedia.org/w/index.php?title=Franziska_Feifalik&oldid=259011659|journal=Wikipedia|language=de}}</ref> * Irma Sztáray — Countess Irma Sztáray de Sztára et Nagymihály (10 July 1863 – 3 September 1940), lady in waiting 1894–1898, and present at Sisi's assassination<ref>{{Cite journal|date=2026-03-23|title=Irma Sztáray|url=https://en.wikipedia.org/w/index.php?title=Irma_Szt%C3%A1ray&oldid=1344904544|journal=Wikipedia|language=en}}</ref> ===Franz Joseph=== ==== Friends and Allies ==== * Albert Edward, Prince of Wales<ref name=":8" /> (244 of 300) * William II, Kaiser of Germany ==== Enemies ==== ==Organizations and Social Networks== === Queens and Political Leaders === * [[Social Victorians/People/Queen Victoria|Queen Victoria]] * [[Social Victorians/People/Sophie of Wurttemberg|Sophie of Württemberg, Queen of the Netherlands]] * [[Social Victorians/People/Mary Todd Lincoln|Mary Todd Lincoln]] * [[Social Victorians/People/Eugenie of France|Empress Eugenie of France]] === Couturiers === * Maison Worth, Paris * Antal Alter, Budapest (still in existence, as Alter és Kiss)<ref name=":4" /> ==Timeline== [[File:Kaiserin Elisabeth von Österreich und Königin von Ungarn.jpg|alt=Old painting of a white woman dressed opulently with jewels and a crown on her head|thumb|Empress Elisabeth of Austria, 1854]] '''1853 August 18''', Franz Joseph I of Austria's birthday, the ball where he chose Elisabeth over her sister Helene, whom Archduchess Sophie had arranged for him to meet. While Helene was dressed and coiffed in a sophisticated way to attract Franz Joseph,<blockquote>Sisi, by contrast, was attired in a simple, pink-and-white cotton tarlatan frock. To give a sense of the disparity between the two sisters’ costumes, cotton tarlatan was what Meg from ''Little Women'' was intending to wear to the neighborhood dance until kind friends took pity on her and lent her a nicer gown.<ref name=":6" /> (84 of 909)</blockquote>'''1854 April 21''', Elisabeth of Bavaria and Franz Joseph I married.<ref name=":4">{{Cite journal|date=2026-01-09|title=Empress Elisabeth of Austria|url=https://en.wikipedia.org/w/index.php?title=Empress_Elisabeth_of_Austria&oldid=1332040784|journal=Wikipedia|language=en}}</ref> Goldstone says, "“Elisabeth’s dress was of shimmering white silk, richly embroidered in gold and silver. The bodice was adorned with white roses, and on her head was perched her mother-in-law’s diamond-and-opal wedding crown, draped in a veil of gold lace."<ref name=":6" /> (216 of 909) '''1855 March 5''', Elisabeth gave birth to Sophie Friederike Dorothea Maria Josepha, named by Archduchess Sophia, her mother in law; Archduchess Sophia generally denied Elisabeth access to Sophie and did not permit her to breastfeed the child.<ref name=":4" /> '''1856 July 12''', Elisabeth gave birth to Gisela Louise Marie, also taken to be raised by Archduchess Sophia.<ref name=":4" /> '''1857 January 5''', Elisabeth and Franz Joseph began a tour of Italy from Venice to Milan, meeting hostility from the people, who were very close to outright rebellion.<ref name=":6" /> (320–322 of 909) '''1857 May 29''', while the family was traveling in Hungary, the two daughters of Elisabeth and Franz Joseph got sick, and Sophie — the youngest one — died, at two years old.<ref name=":1">{{Cite journal|date=2026-01-13|title=Archduchess Sophie of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduchess_Sophie_of_Austria&oldid=1332725850|journal=Wikipedia|language=en}}</ref> '''1857 July 27''', Maximilian (Maximilian I of Mexico) and Charlotte, daughter of Leopold I of Belgium, married.<ref name=":7">{{Cite journal|title=Maximilian I of Mexico|url=https://en.wikipedia.org/w/index.php?title=Maximilian_I_of_Mexico&oldid=1333693281|journal=Wikipedia|date=2026-01-19|language=en}}</ref> Charlotte was Queen Victoria's cousin, so the marriage caused an alliance between the House of Habsburg and the House of Saxe-Corburn and Gotha. Prince Albert and many European royals attended the wedding. '''1858 August 21''', Elisabeth gave birth to Rudolf Franz Karl Josef, the fact that he was male and thus an heir occasioning a 101-gun salute.<ref name=":4" /> '''1860 November 19''', Elisabeth sailed south from Antwerp to Madeira on Queen Victoria's yacht. Somehow Victoria knew how sick she was.<ref name=":6" /> (476 of 909, n. ii) (447 of 909) Her health began to improve immediately, and she ate well on the ship during the rough passage. She had ladies in waiting with her, and as always she was gossiped about maliciously.<ref name=":6" /> (478–482 of 909) [[File:Kaiserin Elisabeth 1862.jpg|alt=Old photograph of a beautiful white woman dressed stylishly|thumb|Elisabeth of Austria, 1862]] '''1861 April 28''', Elisabeth left Madeira to go back to Vienna. She began to be ill again almost immediately and was worse than she had been before she had gone to Madeira. This time, Franz Joseph had his brother Max (Maximilian I of Mexico) take her to Corfu on his yacht.<ref name=":6" /> (482–483 of 909) '''1862''', Sisi had a belt as an exhibit in the Great London Exposition or International Exhibition. David Kunzle, in his 1982 ''Fashion and Fetishism'', says,<blockquote>The pro-corset party called upon an exceedingly prestigious example — that of the beautiful Empress Elizabeth of Austria, reputed to possess the smallest waist ever seen. Together with her portrait, her girdle had been on view at the Great Exhibition. A query as to the exact length of this girdle was answered by Miss Turnour, who, when she visited the Exhibition, had seen the exhibitor hold it up in his hand, “so beautiful, purple velvet stiff with rich gold thread, looking like a dog-collar when clasped.” Miss Turnour was even allowed to make a “frantic and futile effort” to close it round her own slender waist, and was enabled to certify the measurement to be exactly 16 inches; and the Empress was tall, 5 foot 6 inches. The unspoken inference was, of course, that the Empress indulged in tight-lacing, which thereby acquired a social cachet otherwise totally lacking …. ... Why was an object with such scandalous associations put on public display? With her horror of publicity, especially as regards details of her personal life, it seems inexplicable that the Empress would have encouraged gossip around so intimate a matter as a waist-measurement. If the numerous biographies remain silent on this curious episode, is it because domestically the matter was hushed up? After all, in order to protect the imperial dignity the police actively suppressed stories of her equine acrobatics, and destroyed photographs pertaining to it. If the 16 inch belt was displayed with her permission and knowledge (and it seems hard to conceive otherwise) or, worse, on her personal initiative, was it intended as a provocation? Was it the bizarre symbol of or satire upon the exhibitionism to which the most adulated woman in Europe was subject?^{Qtd. in} <ref name=":11" /></blockquote>'''1863 September 3''', Franz Josef and Queen Victoria met at the Ehrenburg Palace (Schloss Ehrenberg) in Coburg, Franconia, what is now Germany.<ref name=":9" /> (499 of 1204) Elisabeth does not seem to have been there, and for Victoria, this was too soon after Albert's death. She had no ministers but Princesses Alice and Helena with her, and she was shy and out of her depth. '''1867''', because of Elisabeth's negotiations, including a decision to have another child and thus to reopen relations with Franz Joseph I, the Austro-Hungarian Compromise of 1867 made Hungary and Austria dual monarchies.<ref name=":4" /> Franz Josef and Elisabeth were crowned king and queen of Hungary, and she set up "her 'own' court in the Royal Palace of Gödöllő, Hungary .... She asked Franz Joseph to finance her a trainer from the Renz Circus and two of its famous equestriennes, Emilie Loiset and Elise Petzold taught her Circus stunts [sic punctuation]."<ref name=":10" /> (1) '''1867 June 19''', Franz Joseph's brother, Maximilian (then Maximilian I of Mexico) was executed by a firing squad of revolutionaries resisting monarchy.<ref name=":7" /> '''1867 December 24''', Elisabeth's 30th birthday, after which she did not permit any photographs to be taken, although paparazzi sometimes succeeded in getting an image.<ref name=":10" /> (4) '''1868 April 22,''' Elisabeth gave birth to Marie Valerie Mathilde Amalie, the "Hungarian child."<ref name=":4" /> '''1869 June''', Franz Joseph and Elisabeth were crowned King and Queen of Hungary.<ref name=":4" /> '''1869 November 15–17''', [[Social Victorians/People/Elisabeth of Austria|Franz Joseph I]] was at the official opening of the Suez Canal.<ref>{{Cite journal|date=2026-01-09|title=Suez Canal|url=https://en.wikipedia.org/w/index.php?title=Suez_Canal&oldid=1331946563|journal=Wikipedia|language=en}}</ref> Was Elisabeth there? '''1872 May 28''', Archduchess Sophia died.<ref>{{Cite journal|title=Princess Sophie of Bavaria|url=https://en.wikipedia.org/w/index.php?title=Princess_Sophie_of_Bavaria&oldid=1333066696|journal=Wikipedia|date=2026-01-15|language=en}}</ref> '''1873''', the Great Exhibition in Vienna attracted people like Albert Edward, Prince of Wales, who "persuaded [Elisabeth] to visit England during the hunting season."<ref name=":10" /> (2) '''1873 April 20''', Elisabeth's daughter Gisela married Prince Leopold of Bavaria. Elisabeth did not attend the wedding.<ref name=":2">{{Cite journal|date=2026-01-13|title=Archduchess Gisela of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduchess_Gisela_of_Austria&oldid=1332726064|journal=Wikipedia|language=en}}</ref> '''1874 January 8''', Gisela's first child was born; Elisabeth attended the christening.<ref name=":2" /> '''1874 August 1''', Elisabeth, her daughter Marie Valerie, Countess Marie Festetics and Elisabeth's entourage, horses and a dog arrived at Ventnor, Steephill Castle, on the Isle of Wight, near Osborne House. (The castle had been rented for 2 months, and some renovations were made before Elisabeth arrived.)<ref name=":10" /> (2) '''1874 August 16''', Queen Victoria visited Elisabeth from Osborne House. Countess Marie Festetics wrote,<blockquote>I was not surprised. [The sight of the Queen] standing next to the Empress was startling, though. Archduchess Valerie was rather scared from the sight. She has never seen anybody with such a strong body. She [Queen Victoria] was quite warm-hearted, though. [interpolations Káli-Rozmis's]<ref name=":10" /> (2)</blockquote>Queen Victoria invited Elisabeth to Osborne House, but Elisabeth declined on the grounds of "delicate health," although she told her mother she declined because she was bored by "that kind of thing."<ref name=":10" /> (2) '''1875''', Elisabeth went to Fecamp, Normandy, France with "her daughter Marie Valerie and a court of 70 people."<ref name=":4" /> '''1876, late winter–early spring''', Elisabeth visited the UK for the hunting season, spending about 6 weeks. '''1876 March''', Elisabeth visited Victoria at Windsor Castle but spent her time with Baron Ferdinand Rothschild.<ref name=":10" /> (2) Victoria did feel these as snubs. '''1876 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there. '''1877, late winter–early spring''', Elisabeth visited the UK for the hunting season with Bay Middleton as her "pilot," spending about 6 weeks. '''1877 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there. '''1878 January''', Elisabeth and her son Rudolph were in the UK, she riding to hounds and he "toured factories and towns, charmed Victoria and her daughter Princess Beatrice, and made the acquaintance [sic; this had already happened] of Edward, Prince of Wales, the queen’s rakish heir."<ref name=":6" /> (760 of 909) On this trip, he believed some untrue gossip that his mother was having an affair with Captain Bay Middleton, who was her "pilot" on the hunting trips; it took her years to forgive him his doubting of her.<ref name=":6" /> (761 of 909) '''1878 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there. '''1879, late winter–early spring''', Elisabeth visited the UK for the hunting season with Bay Middleton as her "pilot," spending about 6 weeks. '''1879 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there. '''1880, late winter–early spring''', Elisabeth visited the UK for the hunting season with Bay Middleton as her "pilot," spending about 6 weeks. '''1881''', Elisabeth purchased a country house in the UK "and had a spiral staircase built from her sitting room into the kitchen, so that she could reach it in private," suggesting some bulimic practices.<ref name=":4" /> (Not clear how this relates to Combermere Abbey, which she rented.) '''1881, late winter–early spring''', Elisabeth rented Combermere Abbey from Colonel Wellington Henry Stapleton-Cotton, 2nd Viscount Combermere.<ref name=":4" /> She was there for the hunting season with Bay Middleton as her "pilot," spending about 6 weeks. '''1881 February 25''', a letter by Ellen Harriet Tollet about Elisabeth hunting:<blockquote>We had a great meet at Woore to see the Empress jump. ... her manner Mrs C says was ‘most queenly.’ She is still a pretty looking woman with a fine figure and an awfully tight habit, so tight, she descended the stairs at Woore Hall sideways, she could not walk straight in her habit. She was ... talking English perfectly. The funniest thing was her enormous orange fan which she used out hunting, when at a check. Where she kept it, I don’t know. She gave £200 to the United Hunt races near Whitchurch yesterday and was present galloping about with her fan up. (''Shropshire Archives'', quoted by Webb) [sic punctuation]<ref name=":10" /> (4)</blockquote>She may have carried the fan to prevent people from seeing her aging face? '''1881 May 10''', Rudolf married Princess Stéphanie of Belgium, a daughter of King Leopold II of Belgium (whose father had married George IV of England's daughter, Charlotte, Princess of Wales).<ref name=":3">{{Cite journal|date=2026-01-13|title=Rudolf, Crown Prince of Austria|url=https://en.wikipedia.org/w/index.php?title=Rudolf,_Crown_Prince_of_Austria&oldid=1332726212|journal=Wikipedia|language=en}}</ref> '''1882, late winter–early spring''', Elisabeth again rented Combermere Abbey from Colonel Wellington Henry Stapleton-Cotton, 2nd Viscount Combermere.<ref name=":4" /> She was there for the hunting season, spending about 6 weeks. '''1882 March''', Elisabeth was in Paris to visit her sister, Duchess Sophie Charlotte in Bavaria.<ref name=":4" /> '''1883''', Elisabeth seems to have given up hunting after the season in 1882. Bay Middleton was getting married and could no longer pilot her, and she had sciatica. Elisabeth said of her giving up hunting, "Suddenly and without all true reason I lost my heart, and I, who did not consider any danger the previous day, then found it in every bush and I could not get rid of its image."<ref name=":10" /> (5) '''1888 March''', Elisabeth was in London, and Bay Middleton visited her there.<ref name=":10" /> (3) '''1888 September''', Albert Edward, Prince of Wales was a guest of Franz Josef and embroiled in a fight with Wilhelm II, whose father Fritz had just died.<ref name=":9" /> (835 of 1204) '''1889 January 30''', Rudolph and his mistress, the 17-year-old Baroness Marie von Vetsera, were found dead of a murder-suicide in his hunting lodge Mayerling.<ref name=":3" /> Elisabeth wore mourning for the rest of her life, black or pearl grey. '''1890 July 31''', Valerie and Franz Salvator married, after her renunciation of rights to the Austrian throne.<ref name=":0" /> '''1892 April 9''', Bay Middleton died in a hunting accident. '''1896-1897''', Elisabeth was staying in the Grand Hôtel du Cap-Martin, in Roquebrune-Cap-Martin, in Provence.<ref name=":4" /> Elisabeth and Victoria met ther in 1896. '''1898 September''', Elisabeth was assassinated by an anarchist, stabbed in the heart by Italian Luigi Lucheni.<ref name=":4" /> ==Demographics== === Nationality === * Elisabeth: Bavarian * Franz Joseph II: Austrian ===Residences=== * 1890: Elisabeth had the palace Achilleion built on Corfu<ref name=":4" /> ==Family== * Elisabeth of Bavaria (24 December 1837 – 10 September 1898)<ref>{{Cite journal|date=2026-01-09|title=Empress Elisabeth of Austria|url=https://en.wikipedia.org/w/index.php?title=Empress_Elisabeth_of_Austria&oldid=1332040784|journal=Wikipedia|language=en}}</ref> * Franz Josef I (18 August 1830 – 21 November 1916)<ref name=":5">{{Cite journal|date=2026-01-22|title=Franz Joseph I|url=https://en.wikipedia.org/w/index.php?title=Franz_Joseph_I&oldid=1334319684|journal=Wikipedia|language=en}}</ref> *# Archduchess Sophie (5 March 1855 – 29 May 1857)<ref name=":1" /> *# Princess Gisela of Bavaria (12 July 1856 – 27 July 1932)<ref name=":2" /> *# Rudolf, Crown Prince of Austria (21 August 1858 – 30 January 1889)<ref name=":3" /> *# Archduchess Valerie (Marie Valerie) (22 April 1868 – 6 September 1924)<ref name=":0" /> ===Relations=== ==Questions and Notes== ==Bibliography== {{reflist}} d6846du14d1r7vwnwcc479xi1qobtc8 0 326987 2807008 2790960 2026-04-29T16:10:43Z ~2026-26080-89 3069312 /* Problem */ 2807008 wikitext text/x-wiki = '''Problem''' = [[File:The Task.png|center|thumb|This is the Task. Be sure to include the modification as an additional question.|429x429px]] Problem Found and Modified by Sarah Jordan '''''<u>Modification</u>''''': Compare your answer for a size n cable with another table’s. Are they the same? How do they compare? ==== Prior Knowledge: ==== As mentioned in the "why my task addresses my standards" section, this type of activity would be given during a quadratic unit. Thus, the students should have worked with quadratic equations recently, and recognize what a quadratic equation looks like. They should also have a little experience in forming a quadratic equation, as in the aspects needed for an equation to be quadratic. === '''Standard(s)''' === A.PAR.6 Build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations. A.PAR.6.2 Fluently choose and produce an equivalent form of a quadratic expression to reveal and explain properties of the quantity represented by the expression. ==== '''Learning goal:''' ==== Students will recognize how a quadratic function can be written in different ways, due to different quantities or ideas represented within the expression, and connect that the different quadratic functions are equivalent. === '''Why my task is high in cognitive demand:''' ===     This task is high-cognitive demand for multiple reasons. First, cognitive effort is required to not only notice the patterns, but to make the patterns themselves (it’s not mindless), and the answer for the general “n” case can be represented in multiple ways, depending on the student’s method of solving the problem. No procedure in the problem is suggested on how to solve it, except for the modification at the end to compare equations to help the problem go better with the standard and my learning goal for the lesson. This means students must notice patterns and build these equations on their own, which may lead to some anxiety when an equation built may not seem to work at first (I experienced this myself multiple times while testing equations I was making based on patterns I noticed). Students are also required to access relevant knowledge to help make sense of the patterns or ideas they notice, and to build the equation, and much self-monitoring if an idea isn’t quite working (I experienced this a couple times as well). (Smith & Stein 2011). === '''Why my task addresses my standard(s):''' ===     This task addresses my standards in a few different ways. First, students are having to “build quadratic expressions and equations…to model real life phenomena”. Though it’s true that students may not recognize initially that their equation is quadratic, '''this type of activity would be given during a quadratic unit''', so hopefully, when they build their equation, and while comparing theirs to others, they would notice that it is a quadratic. This is also a real-life situation of making different sized cables. I also argue that this task also addresses all the standard A.PAR.6.2.  The students are producing a quadratic expression, and with the justify part of the question they will be showing how their equation shows their process/ideas of solving, or, in other words, the students will “reveal and explain properties of the quantity represented by the expression”.  Additionally, with my modification, students will see that they are making “equivalent” forms of the same quadratic expression based on their understanding of the problem. I would also aim to reinforce this idea at the end of the lesson by having students tell me their equations and processes of solving the problem, and then (if this is not already noticed by the students) put the equations in standard form to show this fact. === '''Is this task Motivating?''' ===     As we’ve learned through our reading of Motivation Matters, there are a lot of factors, both internal and external, that can affect a student’s motivation towards a problem, math, or school in general. Motivation can be formed through success, vicarious experiences, is social, lead to long-term attitudes, and so much more (Middleton & Jansen 2018). If a student already enjoys math, is interested in a profession related to cables, or has high self-efficacy, this task may be motivating on its own, without any input from me as the teacher. Though I think some of the motivation to do this task would, hopefully, stem from how I plan to launch the task. The demonstration performed by students and the class discussion of ideas to launch the task will hopefully create an atmosphere in the classroom that gives the student motivation to do the task.       In terms of the task, I would say that it is motivating in a couple different ways, but it depends. As I already mentioned, students who are motivated to do the task because they have a high-self efficacy or want a future career in something related to cables, doing the task and completing it would be motivating. This is not only because they are able to accomplish a high-cognitive demand task, but also because success will help fuel their future motivation. Similarly, students who may gain motivation to do the task with how I plan to launch the task, may also find motivation from the task because they complete a high-cognitive demand task, and are hopefully able to work through it and feel successful from its completion. Though the idea of success in making this task motivating can be said about any task, I think the puzzle quality that this task also has is also motivating. If a student loves puzzles, a task like this with two different pattern solving parts (noticing how cables are built, and a possible solution) may be motivating to that student. = '''Some possible solution methods:''' = Below are some possible solution methods students might have. As mentioned above, the goal of this lesson is for students to realize that two different quadratic functions, though they may initially look different (due to different ways of building the quadratic functions), can be equivalent when you "boil them down" or get them to their standard form. === '''Solving Method 1:''' ===     To solve this problem, I first looked at the photo provided for the 5-size cable, and decoded how the pattern would work for other ones. The longest row on size 5 was nine, which was 2(5)-1, or you add 4 to 5, so you add whatever the size is minus one to the size to get what the longest part of the cable would be (5 +(5-1)=9). Following this pattern, I made a picture equivalent for cables size one through 4. I then counted the total for each one to try and compare them. I noticed that if you subtracted one from each total, you got a number that was equivalent to the size of the cable times a multiple of 3. For example, for size 5, 61-1=60, and 5(12) = 60, and for size 4, 37-1=36, and 4(9) = 36. I then noticed that the multiple of 3 that you multiply the size by is 3 times the size minus 1 (3 x [size-1]). Using the size five example, 12 = 3(4), and 4 = 5-1. This also follows for the case of 4, because 9 = 3(4-1). Thus, with this info, I then built my equation showing this fact. My final equation was (with n being the size of the cable): n(3(n-1))+1. I then used this equation to check that the number I had counted/ estimated for a size ten cable was right, and it was for the answer I got of 271. [[File:Solutions_for_task.png|left|frameless|Work for solutions|418x418px]] [[File:Work for solution 1.png|center|thumb|362x362px]]   [[File:Equation built on ideas.png|center|thumb|368x368px]] [''I then solved my equation to put it in standard form of a quadratic, to compare it to the equation I got for the other method, but this was done after solving it in the other method.''] [[File:Quadratic equation to compare.png|center|thumb]] There were a couple mathematical ideas used in this method. One idea was recognizing patterns, when looking at the size five cable, and figuring out the “rule” for each size so I could build my other sized cables. Additionally, manipulating numbers using multiplication, addition, subtraction, and division, to try and notice/compare how the different sized cables could relate to each other. Also, building a function/equation based on a real-life scenario that represents a certain idea, and recognizing that it was indeed a quadratic equation. === '''Solving Method 2:''' ===     Similarly to the first method, I first looked at the photo provided for the 5-size cable, and decoded how the pattern would work for other ones. The longest row on size 5 was nine, which was 2(5)-1, or you add 4 to 5, so you add whatever the size is minus one to the size to get what the longest part of the cable would be (5 +(5-1)=9). Following this pattern, I made a picture equivalent for cables size one through 4. I then noticed that I could make two trapezoids out of this shape of the cable, one facing upward, and one downwards. I then used the formula for the trapezoid area (h <math>\frac{a+b}{2}</math>) to help build my equation. I knew that a=n (the size of the cable) because that was always the top of the trapezoid, and b= 2n-1, the longest row. Finally, my height was always the same as the size of the cable itself (I counted number of circles high from bottom of the trapezoid to the top) so h=n. There were always two trapezoids that overlapped, so I multiplied the equation I came up with for the trapezoids by 2, and since they overlapped, I subtracted 2n-1 from everything, accounting for the second time the longest row was being counted. My final equation was: 2( <math>\frac{n+(2n-1)}{2}</math>(n))- (2n-1). I then used this equation to check/solve for the 10^th cable, which also gave me 271. I also manipulated this equation into a standard form for a quadratic, to compare it to the equation I got in the first method, and I got the same equation. [[File:Solutions for task.png|center|thumb|419x419px]] [[File:Work for solution 2 and comparing to solution 1.png|center|thumb|464x464px]] There were a couple mathematical ideas used in this method. One idea was recognizing patterns, when looking at the size five cable, and figuring out the “rule” for each size so I could build my other sized cables. Recognizing shapes, like a trapezoid, and recognizing that area could be used to solve this problem. The idea that items should be counted once was also used in this when subtracting the extra row from the equation. Also, building a function/equation based on a real-life scenario that represents a certain idea (in this case using area), and recognizing that it was indeed a quadratic equation. == '''Non-trivial Mathematically Incorrect Solution''' == [[File:Non-trivial Possible incorrect solution.png|center|thumb|571x571px|Non-trivial Possible incorrect solution]]     One non-trivial mathematical incorrect solution may occur during the second solution I proposed to this problem. The student understands how the cables are built, and how the size of the cable relates to how it is built—this is shown by the fact that cables 1-4 are drawn correctly in the student’s solution, and the student sees that the longest row is always 2(n)-1. Secondly, the student correctly sees how the cables can be broken down into two trapezoids, and that the area of a trapezoid can be then used to help solve how many small cables are within each cable. This means that the student understands what area of a shape (in this case a trapezoid) means, which in terms of the problem, is that you can see how much space or cables fill up the two trapezoids, to see the total number of small cables fit into each size cable. Additionally, the student understands what the representation of “n” means, and how it can be used to create a formula to solve for different size cables. However, this is where the misunderstanding occurs. The student’s picture is a little unclear where the two trapezoids land, but the student does say that both trapezoids have “b” in the formula as the longest row, which clears the drawing up, '''but the student misses the idea that, in their formula, the longest row is being counted twice'''. This means that, even though the student understands area, they may misunderstand that it’s a little more than just two trapezoids and may not recognize that even though “a” should be counted twice, that it doesn’t mean that “b” should be counted twice.     Something that I may do if I were to see a student solving a problem in this way, (assuming they seem stuck and confused why the formula isn’t working) is first ask them <u>assessing questions to understand their thinking</u>, even if I already have an idea of their thinking. I would then ask them to point me to the two bases of each trapezoid, or in other words <u>what “a” and “b” represent for each trapezoid</u>. Once has student has done that, I would then walk away for a little bit after asking them the question ''“Should every part of the trapezoid counted twice?”'' The reason I would '''leave is to give them time to think about my question, but also so they wouldn’t look at me when saying their answer to my question'''. After a minute or two, I’d come back and ask them what they found out. Hopefully they can tell me that “a” is counted twice and should be, and that “b” is counted twice but shouldn’t because it’s being used by both trapezoids overlapping; and the longest row is being counted twice instead of once. If they hadn’t already fixed the formula, I would then ask, ''“How can you show that in your formula?”'' and hopefully the student is able to tell me that they just need to subtract the longest side from the answer their original formula gave them. = '''Bell-Ringer (As students are walking in) (About 1-2 minutes of class time)''' = "What is one mathematical idea or concept that you remember us talking about yesterday, and what is one detail about that idea?" * Since this task is taking place during a quadratic unit, the students during this bellringer will most likely answer quadratic functions, or some identifying features of a quadratic function. * The purpose of this warmup is to let the students review to themselves what they learned about the day before, and help get their minds into "math mode" to prepare them for the task ahead. = '''Launching the Task (About 5 minutes)''' =     To introduce my task, one thing that I would do is I would “Discuss the key contextual features of the problem” so I can make sure that my students understand not only the context of the problem, but also ''<u>so they are able to relate more to the problem itself</u>'' (Jackson 2012). It’s quite possible that '''students may or may not know the concept of adding more cables to make a cable stronger while keeping the same shape of the cables'''. What I would do is take a thin piece of string and call two students up to try and break the string. Hopefully, they are easily able to. Then I would give them a piece of string that’s a little thicker. Assuming that it is broken, or even if it wasn’t, I would ask them if it was easier or harder to break. I would assume they would most likely say harder, and I would ask the class why this string is harder to break than the first. I may get a couple different answers to the question like “it’s bigger” or “it’s thicker”, or other answers along those lines. If a student hasn’t already pointed it out, then I would ask the class <u>“What makes the string thicker than the other?”</u> Hopefully, after some class discussion, the class concludes that the thicker string is made up of a bunch of thinner strings put together. With that idea pointed out, I would <u>sum up our “key mathematical ideas” and use the “common language” that the student’s had said and say, “so the thicker string is stronger because it’s made up of a bunch of thinner strings, while keeping the same shape as the first string, as you all have told me”</u> and I’d pull up the problem on the board, and have someone read it for the class. Next, I would tell the class to use our ideas with strings that we had done but instead think of the string as thin metal cables. Even though I would be tempted to point out to them what makes the image a size five cable, to keep the “cognitive demand” of the task, this is the point where I’d set them free to start working on the problem. <u>If someone seemed stuck, I ''may ask them what other size cables might look like to point them in the right direction'', '''but otherwise this is where the launch of the task ends, and the monitoring and supporting of student thinking begins'''.</u> I would tell my students to '''work with their table groups on the problem''', and to '''write down/keep track of any and all ideas''' they have while working on the task, encouraging them to '''get some paper to share, or to write on the problem sheet itself'''. I would remind them to keep track of their ideas, and to answer all of the questions. = '''Monitoring and Supporting Student Thinking (20-30 minutes)''' = === What questions will you ask to understand what the students are doing and to support their progress on the task? === * What does “n+1”, “n+2”, “n+3” represent in your drawing? (''This could be either solution method'') '''(Assessing Question)''' * Why did you line up the sizes and their number of cables like this? (''This is reference to the first solution and the left most part of the sort of table'') '''(Assessing Question)''' * Where is the (n-1) from your equation in your table? (''This is for the first solution'') '''(Assessing Question)''' * So, I see that you’ve subtracted 1 from each total, how come? '''(Assessing Question)''' OR Is there a way you could convert 61, 37, 19, 7, and 1 to easier numbers to work with? (''Also, for first solution'') '''(Advancing Question)''' * Can you represent your ideas in the table [or from the drawing of the trapezoids] in an equation? (''both solutions'') ('''Advancing Question'''/ relating back to the prompt) * Where did 2n-1 come from? (''Second solution'') '''(Assessing Question)''' * I see you’ve drawn two trapezoids from your drawings of the different size cables, what are you going to do with that? (''Second solution'') '''(Assessing Question)''' * What will the other size cables look like? (''Could be used for either solution if the student(s) seem stuck on where to start)'''''(Advancing Question)''' === Identify places where these students might get stuck. How will you respond to them? === * '''If a student seems to be on the path of solution method 1, and has written the number of the cable next to their totals, and seems stuck on how to relate their numbers''' I would ask them “Is there a way you could convert 61, 37, 19, 7, and 1 to easier numbers to work with?”  Or if they have gotten to the point of noticing they could subtract one, and now have 60, 36, etc, and have the 5x12, 4x9, etc, but seem stuck on how to make their findings into a formula, I’d probably say something along the lines of  “So based off of the problem, we need to make an equation to find the nth cable. This means that we need to have an equation based around the variable n. Looking at the steps in your table, how can you relate those ideas to ‘n’ which represents the size of the cable?” I would leave the student at this point to let them think about my question, and if I come back and they still are having difficulty building the equation (like maybe they have n(     )+1) Then I would say, “I see that you have the start of an equation, can you explain where each part of your equation is shown in your table? (student responds and shows) Okay, yeah, I can see that. So, you haven’t represented the 12, 9, 6, 3 parts in your equation yet. What do you notice about those numbers and how can n be used to represent those ideas?” * '''Similarly to solution method one, in solution method 2,''' '''it’s possible a student may get stuck on forming their equation. If they seem to be stuck on the part where they are identifying what ‘a’ and ‘b’ in terms of n''', then after asking them some questions to understand their thinking (like what they’ve realized so far) then I would say to them, “Well, looking at the equation that you have written for the area of a trapezoid, can you point to what each letter in your equation represents in your picture?” Hopefully, the student would point and then realize that ‘a’ & ‘h’ would be n, and ‘b’ would be 2n-1. If they don’t realize after a bit of time, then I may ask, “So, with what you pointed to, is there a way you could express that in terms of n?” Another part a student may get stuck on may be like the non-trivial solution, as in they may not recognize to subtract the repeated part, or they may know they need to, but they may be stuck on where they should subtract it. If it’s the second one (since I’ve previously described what I’d do for the first one) then I would ask them, “So you’re saying that we need to subtract the repeated long row. Where are you thinking you should subtract it from?” If they seem stuck on it being inside the parentheses or outside, I will ask, “Does it matter where it’s subtracted from? (Student’s response) Oh so you’re thinking it should be outside? How come?” Hopefully they’d respond with something along the likes of ‘if I do it in parentheses it would get rid of b or be counted twice, so it should be outside’. === What are some things you will <u>be sure '''''not''''' to ask or say</u>? === * What if you subtracted one from the totals of each size? (''Solution 1'') * What times 3 gets you 9, what times 3 gets you 6, what times 3 gets you 12? How do those answers relate to the size of the cable? ''(Solution 1, I feel like even though the second part of the question seems okay, the first part is leading the student to a given answer instead of having them come up with it themselves.'') * This long part is counted twice, so you need to subtract it from the total, so you aren’t over counting. (''Solution 2'') * Draw trapezoids. (''Solution 2'') * The longest part of the cable should be 2n-1. (''Either solution''.) * Make a table comparing the totals of each cable size. (''Solution 1'') === What if a group finishes quickly? === It is possible that students may finish the task early. * If only one group has, then I will ask them to come up with another solution using a different strategy than they had previously. * If two groups have, I would then have them move to the modification part (which every table has) and have them compare and discuss their ideas. ** If they finish their discussion before all the other tables have gotten the chance to discuss, I would either push their discussion a little more (like if they haven't discovered the same standard equation part that I want them to see, then I may ask them some questions similar to ones I would do in the final discussion) or I would again ask both tables to work together to see if they can come up with a second (if they did the same method) or third solution, until the other tables have a chance to discuss. == '''Making Connections and Addressing the Learning Goal'''  '''(10-15 minutes)''' ==     There are a couple mathematical connections between my two solution methods. The first one actually relates to my learning goal, since <u>both the equation for solution 1</u>, which is n(3(n-1)+1, <u>and the equation for solution 2</u>, which is 2( <math>\frac{n+(2n-1)}{2}</math>(n))- (2n-1), <u>both represent the same quadratic function which can be seen when both expressions are written in standard form</u>: 3n²-3n+1. Another mathematical connection between the two solutions is that <u>both solutions require an understanding of the pattern of how the cables are built</u>. Both solutions have drawings of cables 1-4 based on the size 5 cable given. This means the identification of what makes a cable that size was discovered and understood, to get towards a solution. To be specific, each cable has the same number of smaller cables as its size for the top, bottom, and all the sides of the hexagon cable.       My learning goal for this task is to show my students '''how an algebraic/numerical type solution (solution 1) and a geometric solution (solution 2) or any other ‘correct solution’ can all have equations that look different, but represent the same quadratic expression''', which can be seen when re-written in standard form of a quadratic expression. I lead to this idea by modifying my task, and I show how the equations boil down to the same equation in both my solutions. To accomplish this, I would spend the '''''<u>last 10-15 minutes of class</u>''''', bringing the students back together after all the tables have been able to finish this task (hopefully most students have at this point). <u>I would ask a table that I’d seen solution method one in to share their equation, and I’d ask another table to share their equation (the second solution method). I would also ask if any other table had a different equation that worked. I would have each group share their method and equations to the class.</u> '''Once I had the different equations on the board, I’d ask my students if there was any way to compare their equations.''' After there is some discussion, having students explain how they got their equation and listening to their posed ideas on how to compare them, I may steer the discussion towards the standard form of a quadratic (if not already done so) by asking <u>“What type of function or relationship are these equations?”</u> If someone answers quadratic, I’d ask them why. After they explain to me why, I would say, “If these are quadratic, could we put them in the standard form for a quadratic? (students say yes or nod their heads) Let’s try that!” '''Once we realize that all the different equations look the exact same in standard form, I’d say to the students, “Oh, so even though we had three different looking equations which represented different ideas based off of you all’s solutions, they all are the same expression.”''' == '''Check Student Understanding/ Exit Ticket (About 2-4 minutes)''' == "In your own words, what did we learn today about quadratic functions?" * This provides the students a chance to reflect on the point of the lesson, and provides us teachers a chance to see what students took away from the lesson. * Though there is a hopeful correct answer, there is technically no right answer for this question, as the students are asked to put their answer into their own words. * The idea is to review these exit tickets to determine what the next day's warmup should be, based on student understanding. = '''Sources/References''' = Jackson, K. J., Shahan, E. C., Gibbons, L. K., & Cobb, P. A. (2012). Launching complex tasks ''Mathematics Teaching in the Middle School'', ''18''(1), 24- 29. <nowiki>https://doi.org/10.5951/mathteacmiddscho.18.1.0024</nowiki> Middleton, J. A., & Jansen, A. (2011). ''Motivation matters and interest counts: Fostering engagement in mathematics.'' National Council of Teachers of Mathematics. Smith, M. S., & Stein, M. K. (2018). ''5 practices for orchestrating productive mathematics discussions''. National Council of Teachers of Mathematics. == '''Source of Problem''' == ''Steel Cables, NRICH''. (2025). Maths.org. <nowiki>https://nrich.maths.org/problems/steel-</nowiki> cables?tab=overview [[Category:Lesson plans created by prospective teachers]] n7wbgunk1cu95ythlvq2s912l7v79fx 0 326988 2807015 2794434 2026-04-29T16:14:45Z Esk297 3044098 Changed the Title 2807015 wikitext text/x-wiki = Quadratic Expressions and the Handshake Problem: A Task Plan = = Preparation for Task Implementation = == Georgia Standards: == A.PAR.6: Build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations. * A.PAR.6.1 Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context. * A.PAR.6.3 Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena. == Learning Goals: == * Given a real-life quadratic problem, students will build, interpret, and solve a quadratic expression in context. * Students will connect arithmetic equations to quadratic equations by recognizing that while the rate of change of arithmetic equations is constant, the rate of change of the rate of change of quadratic equations is constant. (''done through sequencing; need a student to make a table, though extremely likely with the first and second questions'') == Class Layout: == For this class it is ideal if you are able to arrange the desks into groups of 5 or have the students in groups of 5 at whitboards. If so you should make sure that there is space for the groups to stand. == Materials: == * Handout * Graph Paper (Only by request) ''I don't want to push students towards that solution strategy, but I'm happy if they come to it on their own.'' * Graphing Software (by request) ''Same reasoning.'' * Calculators == Launch (10-20 minutes): == Ask the students: What makes a good handshake? In this question I am attempting to solicit input from my students since Jackson et al. argues, “In the effective launches that we identified, teachers did not simply talk to students about the key features of tasks but instead solicited input from their students” (2012) (''I don’t love the question since it feels off topic but I couldn’t think of another good question about handshakes)''. After fielding the responses from the introductory question, ask the class ‘when you give someone a handshake, someone else is also giving you a handshake. How many handshakes are occurring? Why? This discussion should help thwart the issue of double counting handshakes from the incorrect solution and support students’ understanding. After we conclude this discussion we look towards success criteria/goals and expectations. Tell the students, "As we work through this problem, our goals are not just about the math you are working on, but also being able to communicate your ideas with your group. Also, I will be coming around asking why you chose a strategy, and one of our goals is to be able to explain it. Remember, as usual, everyone should get a chance to contribute to the group, and we will respect everyone's contributions." Once I have the expectations set and I believe that the students have a good grasp on how to count handshakes, ask them to get into groups of 5, shake everyone’s hand exactly once, and count how many handshakes occurred for the entire group. Once they have started that I put the rest of the questions on the board/projector (or give them a worksheet) and start walking around the room helping as needed. This setup doesn’t suggest a method which is important in order to maintain the cognitive demand (Jackson, 2012). == The Problem (40-50 minutes): == 1.  Get in a group of 5 and shake everyone else’s hand, but shake each person’s hand only once.  How many handshakes were there for the whole group? ''(I changed this a little to make it more motivational)'' 2.  If a 6th person joins your group, how many handshakes will they give? (If the groups are not even, you can change this to another.) 3. If everyone in class shakes everyone else’s hand, how many handshakes would there be? (Sample class size: 30 people) (''changed class size so they had even groups of 5, but it should work for any class size.)'' 4. If there were 45 handshakes, how many people would shake hands? (''This is to hopefully lay some groundwork for later exploration when we move to factoring and the quadratic equation.) (I deleted the one about the full school, I think it was just excessive and wasn’t productive.)'' 5. Create a solution guide. Include step-by-step walkthroughs and explanations. (''I really liked some of the student creations with this. I think it helps them wrap everything up for themselves, which I believe is helpful.)'' == Peer Audit and Quick Revision (5 minutes): == Once every group has made a poster I will have them send an auditor to another group. The auditor's job is to see if they can follow the other group's poster. I will stress that they are not meant to critique the math, instead they are meant to focus how understandable the poster is. If they can't follow it well they should explain where they got stuck and then the group can work to clarify that portion. == Whole Class Discussion (15 minutes): == For the sequencing I would have a group who chose the incorrect solution strategy go first. I would ask them to share their poster and walk the class through their mathematical streatgy and their reasoning. Then open the floor to ideas and thoughts from the class and let them discuss. Then I would move towards Solution 2 and do the exact same thing with them. Then Solution 1. This order should allow the group with the incorrect solution strategy get some experience trying to explain their solution strategy and give the whole class practice debating and refuting ideas. A student may just say, "You just counted every handshake twice!" after that I would ask them to explain the statement and then the presenter a chance to refute it. Then I may explain that this just hit the success criteria since the group communicated so effectively that the other students could understand their process and that they made a pattern and generalized it. If no-one speaks up after I have the student present the incorrect solution I can ask, "Group A says there are 20 handshakes for 5 people, but when we did it physically in our groups, we counted 10. Can someone help us figure out where those extra 10 came from in Group A's logic?" After we do the same for the other 2 solutions and and other ones I think should present, we can use Solution 1 to try and generalize the handshake problem into a formal expression. ==== What specific questions will you ask so that students will— ==== # make sense of the mathematical ideas that you want them to learn? # expand on, debate, and question the solutions being shared? # make connections among the different strategies that are presented? # look for patterns? # begin to form generalizations? 1) Why does the number of handshakes increase by a different amount for each person added? 2) Group A says there are 20 handshakes for 5 people, but when we did it physically in our groups, we counted 10. Can someone help us figure out where those extra 10 came from in Group A's logic? 3) What does the <math>n-1</math> mean in these solutions? How does it connect to the graph? 4) What do we notice about the change in handshakes for each added person? 5) Could we use this same logic to find the handshakes for a group of 100? What would the formula look like then? == Building on this class: == The next day we would work through what makes a quadratic equation now that they have seen one and have a representation. They would work on the change of the change being constant and how we can see it in our tables and graphs from the handshake problem. == Success Criteria: == In this task success has several components: * I can communicate mathematical ideas with peers * I can explain my method of solving a problem * I can identify a pattern and generalize it into a formal relationship == Reasoning: == First, the context of when I am using this task is introducing quadratic equations in Algebra 1. This task begins with procedures with connections in the first four questions, since the procedure isn’t given, and it is focused on developing an understanding of the concepts. Then we go to the 4th question, which requires a much higher level of thinking since I hadn’t yet given them the quadratic formula and they hadn’t worked with factoring, but it is meant as an introduction to the topic (I am curious if this is too far). Since it has no suggested algorithm, it explores relationships and requires significant effort; question 4 pushes it into doing mathematics. Then, question 5 asks the students to make a solution guide, which leads them to explore the connections further, wrapping everything into doing mathematics. (Smith & Stein, 2018) == Why this task fits the standards and the learning goal: == The problem presents a real-life phenomenon that is modeled by a quadratic equation. It begins with a fairly low floor, having students plot a couple of situations, then forcing them to build a quadratic equation for question 3. Part 4 asks students to solve their quadratic equations in a mathematically applicable situation. The problem easily meets the first performance goal, as in the first four questions, the students build and interpret the scenario to create a quadratic equation, then ask the student to solve it in question 4. The second goal is far harder and will mostly fall to monitoring and selecting students who noticed the rate of change. Question 2 will hopefully lead some students to that idea, so I can open that discussion at the end of class. If it hasn’t, I would ask some students questions to help them come across this change. I am hopeful they would consider this rate of change, since, theoretically, it would come from a unit or at least a day of arithmetic. == Motivations == This task can be motivational because it begins with a social and kinesthetic activity: students physically interact, forming groups and shaking hands. This task contextualizes the problem from the start, embedding the problem within an everyday action. This does face a struggle since the connection isn’t particularly meaningful. Middleton and Jansen assert, “Contexts can promote personal investment in mathematics only if they are personally relevant and meaningful for students” (2011). While it isn’t incredibly meaningful, some motivation can be made up because this task is challenging and interesting. Since the problem requires a level of pattern recognition, it can lead to aha moments from students when they see the connection between the handshakes and the quadratic expression. These moments, associated with success at a challenging task, bolsters students’ competence, allowing them to “reap the good feelings associated with competence” ​​(Middleton & Jansen, 2011). = Anticipating Student Solutions = == Solution 1. == === Parts 1 and 2. (This student added up each summation to create the equations, then leveraged the pattern they saw to realize that it was <math display="inline">\frac{n(n-1)}{2}</math>, an important step. It is also helpful that they used the abstract form of the summation. ) === {| class="wikitable" |1 |0 |<math>0=\frac{0n}{2}=\frac{n(n-1)}{2}</math> |- |2 |1 |<math>(n-1)=\frac{1n}{2}=\frac{n(n-1)}{2}</math> |- |3 |3 |<math>(n-1)+(n-2)=\frac{2n}{2}=\frac{n(n-1)}{2}</math> |- |4 |6 |<math>(n-1)+(n-2)+(n-3)=3n-6=\frac{3n}{2}=\frac{n(n-1)}{2}</math> |- |5 |10 |<math>(n-1)+(n-2)+(n-3)+(n-4)=4n-10=\frac{4n}{2}=\frac{n(n-1)}{2}</math> |- |6 |15 |<math>(n-1)+(n-2)+(n-3)+(n-4)+(n-5)=5n-15=\frac{5n}{2}=\frac{n(n-1)}{2}</math> |} '''5 people require 10 handshakes, and the 6th person shakes 5 hands.''' === Part 3. (For part 3 they just plugged the numbers into their equation. They showed that they knew how to plug the inputs into their equations. ) === <math>\frac{30(30-1)}{2}=435 \ handshakes</math> === Part 4. (In part 5 they realized that they needed 2 consecutive numbers to equal 90 so they used guess and check to find the answer. ) === <math>45=\frac{n(n-1)}{2}</math> <math>90=n(n-1)</math> Must be 2 consecutive numbers; guess and check {| class="wikitable" |4 |5 |20 |- |7 |8 |56 |- |12 |13 |156 |- |10 |11 |110 |- |8 |9 |72 |- |9 |10 |90 |} n=10 == Solution 2. == === Part 1. (In part one the student thinks person by person using a table. ) === {| class="wikitable" |E | | | | |X |- |D | | | |X |1 |- |C | | |X |1 |2 |- |B | |X |1 |2 |3 |- |A |X |1 |2 |3 |4 |- | |A |B |C |D |E |} With 5 people, person A shakes 4 new hands, B then shakes 3 new hands, C then shakes 2 new hands, then D shakes 1 new hand, E has already shaken everyone’s hand so they have no new hands to shake so there have been <math>4+3+2+1=10 \ handshakes</math> === Part 2. (The student shows an understanding of how the increase affects the count with the addition of another person.) === {| class="wikitable" |F | | | | | |X |- |E | | | | |X |1 |- |D | | | |X |1 |2 |- |C | | |X |1 |2 |3 |- |B | |X |1 |2 |3 |4 |- |A |X |1 |2 |3 |4 |5 |- | |A |B |C |D |E |F |} With 6 people there are <math>5+4+3+2+1=15 \ handshakes</math> so then there is just 1 more handshake for each of the previous 5 people. '''Therefore the 6th person shakes 5 total hands.''' By plotting these points, we can see a quadratic relationship. Since we know it is a quadratic equation, we can start with 3 people; <math>y=32</math> where everyone shakes two hands and their own Then, since everyone doesn’t shake their own hand we can remove those 3 handshakes Then we are still double counting each handshake so we must divide by 2 to only count each handshake once For part 3 they used a graphical representation of the quadratic expression and a vertical line to find the number of handshakes for each problem. This showed a good understanding of intercepts where they realized that they could use a line to find the y at a certain x. === Part 3. === <math>\frac{30^2-30}2=435 \ handshakes</math> === Part 4. === The student uses a graph and a horizontal line to find when there are 45 handshakes, they showed great understanding of intercepts. They also showed that they understood that the answer could only be positive in the context of this probem. Possible solution 3 that I want to stifle is using the summation in a calculator to easily get the answer because it doesn’t require students to understand the quadratic nature of the problem. == Incorrect Solution == === Part 1 and 2. === The premise behind this solution is that the students working out thought about each person shaking hands and said person 1 shook everyone else’s hand and then person 2 did the same and so on. {| class="wikitable" |1 |0 |<math>0=0n=\frac{n(n-1)}{2}</math> |- |2 |2 |<math>(n-1)+(n-1)=n(n-1)</math> |- |3 |6 |<math display="inline">(n-1)+(n-1)+(n-1)=3(n-1)=n(n-1) \ (each\ person\ shakes\ n-1\ hands)</math> |- |4 |12 |<math>(n-1)+(n-1)+(n-1)+(n-1)=4(n-1)=n(n-1) </math> |- |5 |20 |<math>(n-1)+(n-1)+(n-1)+(n-1)+(n-1)=5(n-1)=n(n-1)</math> |- |6 |30 |<math>(n-1)+(n-1)+(n-1)+(n-1)+(n-1)+(n-1)=5(n-1)=n(n-1)</math> |} The first 5 people shake 20 hands and the sixth adds 10 handshakes. === Part 3. (For part 3 and 4 they just plugged the numbers into their equation. They showed that they knew how to plug the inputs into their equations. ) === 30(30-1)=870 handshakes === Part 5. (In part 5 they knew the factors of 45 and realized that they didn’t have 2 consecutive factors so they declared that it was impossible. ) === 45=n(n-1) Consecutive Factors of 45 {| class="wikitable" |45 |5 |9 |- |45 |3 |15 |- |45 |1 |45 |} Since these are the only factors of 45 and none of them are consecutive, 45 handshakes is impossible. Redirection. To redirect this student I would point them back to when they added the sixth person. I would ask, “when you added your sixth person how many handshakes did you add” They would likely say 10 since that is what they counted. Then I would ask, “Can you count out the handshakes?” If needed I can bring it back into the group and have them act it out and have the struggling student count it out, aloud. Then once they realize it is 5 give them time to think it out and find their mistake on their own. If they are still struggling to bring everything together have them draw out each of the situations (1-5 people) on their paper. = Monitoring and Supporting Student Thinking = === Questions to support progress === Working in groups counting up the handshakes for the first 5 people: '''Assessing''': How are you keeping track of the handshakes to make sure you don’t miss any? '''Advancing''': How might you represent the handshakes? (Leaving it open to various representations, but still advancing them towards a representation.) '''Don’t Say''': Can you make me a graph of these results. (Forces the students towards one representation.) When building a table in solution strategy 1: '''Assessing''': Have you noticed any patterns when you’re counting handshakes? How are you representing the patterns? '''Advancing''': Can you predict how many handshakes would be added for the 7th person? Can you make a formula to predict how many handshakes there would be for any number of people? '''Don’t Say''': "Isn't it just n times n minus 1 divided by 2?" '''If they are struggling to notice the pattern''': Let’s break it down, how many handshakes does each person give? '''Focusing''': If I wanted to find how many handshakes after the 12th person, how would I go about doing that? Then ask hows and whys to try and focus the student understanding. This is based on Herbel-Eisenmann & Breyfogle’s paper, ''Questioning our patterns of Questioning.'' (2005) Once the students have the formula: '''Assessing''': Why do you divide by 2? '''Advancing''': Can you explain the formula within the context of handshakes? When trying to work backwards during part 4 in strategy 1: '''Assessing''': What are you trying to find in this problem? How is this different from the other parts? What information do we know? What strategies are you trying to use in this problem? '''Advancing''': What relationship are you seeing between number of people and number of handshakes? '''Funneling''': How does the formula we found relate to this problem? Can you set up an equation with what you have? '''If students struggle''': It’s ok if this takes a few tries, keep thinking about how the number of handshakes relates to the number of people. Let’s break it down, what information do we know and what are we tring to find? Do we know anything else? From there where do you think we should go? '''What I won’t say''': You need to use the quadratic formula. (we haven’t learned it, and even if we did, it would eliminate the struggle) When using a graphical representation in solution strategy 2 parts 3 and 4: '''Assessing''': What results are you getting from your graph? '''Advancing''': Do the results make sense within the problem? How many results are you getting? Why do you get multiple results? '''If they are struggling to figure out whether to use the positive or negative result''': Within the problem which of these answers makes sense? == What to do for fast finishers?: == If there are fast finishers I will first make sure that their poster is shows their solution strategy well, otherwise they should spend more time on their poster. If the poster is good though I will ask them, "Can you find another solution strategy?", and then once they have I can ask them to make another poster. If multiple groups finish quickly, I can have them send an auditor to another group that is finished who's job is to see if they can follow the other group's poster. I will stress that they are not meant to critique the math, instead they are meant to focus how understandable the poster is. If they can't follow it well they should explain where they got stuck and then the group can work to clarify that portion. There can be multiple rounds of this auditing. == What to do for students just working on the poster?: == The poster is at the end of the task and so unless the student is working on it before they have finished the other problems then I am happy with them working on it until we run out of time. If they seem to focus too much on the artistic stide without showing the math, I can ask, "Can you explain how this poster is showing your solution and explanation?" For all I know they may have a great explanation/plan for how their artistic portion will tie into their solution and I wouldn't want to prevent that. For students working on the poster while working on the other parts I'm fine with that unless they are just doodling before they finish their math. Again I can ask, "Can you explain how this poster is showing your solution and explanation?" and they may have a good explanation. If they don't I can ask them to finish their solution strategy before they work on the poster. = Making Connections and Addressing Learning Goal = The chart above visually represents the mathematical connections between the two student solutions to the handshake problem. Solution 1 uses an Input-Output Table to establish a numerical relationship between the number of people and the corresponding number of handshakes. This table serves as a foundation for identifying the quadratic pattern in the data. By observing the differences in the number of handshakes as the number of people increases, students are able to generalize this pattern into the algebraic expression n(n-1)2​, which I labeled in the chart as the Generalized Quadratic Expression. This process highlights the connection between numerical patterns and their algebraic representations, a key learning goal of this task. Solution 2, on the other hand, employs a Two-Way Table that organizes the new handshakes per person. The table gives the points required for the graphical representation of the quadratic function. This Visualizing Quadratic Function stage demonstrates how the results from the table can be translated into a graphical representation, allowing students to visualize the parabolic curve that represents the handshake relationship. This connection between numerical and graphical representations provides a deeper understanding of the quadratic function and its behavior. The chart also emphasizes the equivalence between the two forms of the quadratic expression, n(n-1)2 and x2-x2​. The Equivalence arrow highlights that different representations can express the same mathematical relationship, further reinforcing the connection between algebraic and geometric solutions. These connections directly align with our learning goals. Firstly, students are expected to build, interpret, and solve a quadratic expression in context. Solution 1 demonstrates the building of the quadratic expression from numerical data. The solution shows how you can use a summation using the input to build a quadratic expression, while Solution 2 focuses on the identifiers in the graphical representation allowing you to realize that it is quadratic. Then Solution 2 builds the expression with the knowledge that it is quadratic. Both solutions ultimately lead to the same algebraic result, showcasing the interconnectivity of these representations. Secondly, students are expected to connect arithmetic equations to quadratic equations by recognizing that while the rate of change of arithmetic equations is constant, the rate of change of the rate of change of quadratic equations is constant. Solution 1 demonstrates this well when you point out that since the output is growing by n-1 each time the nout increases by 1. This shows that the rate of change is increasing by 1 each time. I woul use both of these solutions in group discussions since they both “lead to generalized rules” (Smith & Stein, 2018). == Feedback == A.PAR.6.3: Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena. * I don't think students are likely solve this by creating an equation in this context. I think that this task does a good job of the first part, and this is a precursor of the second. Your learning goal is about comparing with arithmetic which isn't quite what the standard and task is about directly. The fully aligned goal is your first one, but it is more of a performance goal and not so much about what students will learn about. Very thoughtful launch! Solution 1: * Understandings are present but not appropriately specific AND not sufficiently connected to the solution method. == References == Georgia Department of Education. (n.d.). ''GaDOE SuitCASE for Mathematics''. <nowiki>https://case.georgiastandards.org/e9dd7229-3558-4df2-85c6-57b8938f6180/e9dd7229-3558-4df2-85c6-57b8938f6180</nowiki> Herbel-Eisenmann, B. A., & Breyfogle, M. L. (2005). Questioning our patterns of questioning. ''Mathematics Teaching in the Middle School'', ''10''(9), 484-489. Jackson, K., Shahan, E., Gibbons, L., & Cobb, P. (2012). Launching complex tasks. ''Journal of Mathematics Teacher Education'', ''15''(3), 223-242. Middleton, J. A., & Jansen, A. (2011). ''Motivation matters and interest counts: Fostering engagement in mathematics''. National Council of Teachers of Mathematics. ¹ Miller, B. ''The handshake problem''. mrmillermath. (n.d.). <nowiki>https://www.mrmillermath.com/2014/02/06/the-handshake-problem/</nowiki> Smith, M. S., & Stein, M. K. (2018). ''5 practices for orchestrating productive mathematics discussions.'' National Council of Teachers of Mathematics. Warshauer, H. K. (2015). Strategies to support productive struggle. ''Mathematics Teaching in the Middle School'', ''20''(7), 390-393. <nowiki>https://doi.org/10.5951/mathteacmiddscho.20.7.039</nowiki> [[Category:Lesson plans created by prospective teachers]] emufic0vsgahlsvvg5xvx1bqu64mmky 2807022 2807015 2026-04-29T16:16:48Z Esk297 3044098 Changed the Display Title 2807022 wikitext text/x-wiki {{DISPLAYTITLE:Lesson plans created by prospective teachers/Quadratic Expressions and the Handshake Problem: A Task Plan}} = Quadratic Expressions and the Handshake Problem: A Task Plan = = Preparation for Task Implementation = == Georgia Standards: == A.PAR.6: Build quadratic expressions and equations to represent and model real-life phenomena; solve quadratic equations in mathematically applicable situations. * A.PAR.6.1 Interpret quadratic expressions and parts of a quadratic expression that represent a quantity in terms of its context. * A.PAR.6.3 Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena. == Learning Goals: == * Given a real-life quadratic problem, students will build, interpret, and solve a quadratic expression in context. * Students will connect arithmetic equations to quadratic equations by recognizing that while the rate of change of arithmetic equations is constant, the rate of change of the rate of change of quadratic equations is constant. (''done through sequencing; need a student to make a table, though extremely likely with the first and second questions'') == Class Layout: == For this class it is ideal if you are able to arrange the desks into groups of 5 or have the students in groups of 5 at whitboards. If so you should make sure that there is space for the groups to stand. == Materials: == * Handout * Graph Paper (Only by request) ''I don't want to push students towards that solution strategy, but I'm happy if they come to it on their own.'' * Graphing Software (by request) ''Same reasoning.'' * Calculators == Launch (10-20 minutes): == Ask the students: What makes a good handshake? In this question I am attempting to solicit input from my students since Jackson et al. argues, “In the effective launches that we identified, teachers did not simply talk to students about the key features of tasks but instead solicited input from their students” (2012) (''I don’t love the question since it feels off topic but I couldn’t think of another good question about handshakes)''. After fielding the responses from the introductory question, ask the class ‘when you give someone a handshake, someone else is also giving you a handshake. How many handshakes are occurring? Why? This discussion should help thwart the issue of double counting handshakes from the incorrect solution and support students’ understanding. After we conclude this discussion we look towards success criteria/goals and expectations. Tell the students, "As we work through this problem, our goals are not just about the math you are working on, but also being able to communicate your ideas with your group. Also, I will be coming around asking why you chose a strategy, and one of our goals is to be able to explain it. Remember, as usual, everyone should get a chance to contribute to the group, and we will respect everyone's contributions." Once I have the expectations set and I believe that the students have a good grasp on how to count handshakes, ask them to get into groups of 5, shake everyone’s hand exactly once, and count how many handshakes occurred for the entire group. Once they have started that I put the rest of the questions on the board/projector (or give them a worksheet) and start walking around the room helping as needed. This setup doesn’t suggest a method which is important in order to maintain the cognitive demand (Jackson, 2012). == The Problem (40-50 minutes): == 1.  Get in a group of 5 and shake everyone else’s hand, but shake each person’s hand only once.  How many handshakes were there for the whole group? ''(I changed this a little to make it more motivational)'' 2.  If a 6th person joins your group, how many handshakes will they give? (If the groups are not even, you can change this to another.) 3. If everyone in class shakes everyone else’s hand, how many handshakes would there be? (Sample class size: 30 people) (''changed class size so they had even groups of 5, but it should work for any class size.)'' 4. If there were 45 handshakes, how many people would shake hands? (''This is to hopefully lay some groundwork for later exploration when we move to factoring and the quadratic equation.) (I deleted the one about the full school, I think it was just excessive and wasn’t productive.)'' 5. Create a solution guide. Include step-by-step walkthroughs and explanations. (''I really liked some of the student creations with this. I think it helps them wrap everything up for themselves, which I believe is helpful.)'' == Peer Audit and Quick Revision (5 minutes): == Once every group has made a poster I will have them send an auditor to another group. The auditor's job is to see if they can follow the other group's poster. I will stress that they are not meant to critique the math, instead they are meant to focus how understandable the poster is. If they can't follow it well they should explain where they got stuck and then the group can work to clarify that portion. == Whole Class Discussion (15 minutes): == For the sequencing I would have a group who chose the incorrect solution strategy go first. I would ask them to share their poster and walk the class through their mathematical streatgy and their reasoning. Then open the floor to ideas and thoughts from the class and let them discuss. Then I would move towards Solution 2 and do the exact same thing with them. Then Solution 1. This order should allow the group with the incorrect solution strategy get some experience trying to explain their solution strategy and give the whole class practice debating and refuting ideas. A student may just say, "You just counted every handshake twice!" after that I would ask them to explain the statement and then the presenter a chance to refute it. Then I may explain that this just hit the success criteria since the group communicated so effectively that the other students could understand their process and that they made a pattern and generalized it. If no-one speaks up after I have the student present the incorrect solution I can ask, "Group A says there are 20 handshakes for 5 people, but when we did it physically in our groups, we counted 10. Can someone help us figure out where those extra 10 came from in Group A's logic?" After we do the same for the other 2 solutions and and other ones I think should present, we can use Solution 1 to try and generalize the handshake problem into a formal expression. ==== What specific questions will you ask so that students will— ==== # make sense of the mathematical ideas that you want them to learn? # expand on, debate, and question the solutions being shared? # make connections among the different strategies that are presented? # look for patterns? # begin to form generalizations? 1) Why does the number of handshakes increase by a different amount for each person added? 2) Group A says there are 20 handshakes for 5 people, but when we did it physically in our groups, we counted 10. Can someone help us figure out where those extra 10 came from in Group A's logic? 3) What does the <math>n-1</math> mean in these solutions? How does it connect to the graph? 4) What do we notice about the change in handshakes for each added person? 5) Could we use this same logic to find the handshakes for a group of 100? What would the formula look like then? == Building on this class: == The next day we would work through what makes a quadratic equation now that they have seen one and have a representation. They would work on the change of the change being constant and how we can see it in our tables and graphs from the handshake problem. == Success Criteria: == In this task success has several components: * I can communicate mathematical ideas with peers * I can explain my method of solving a problem * I can identify a pattern and generalize it into a formal relationship == Reasoning: == First, the context of when I am using this task is introducing quadratic equations in Algebra 1. This task begins with procedures with connections in the first four questions, since the procedure isn’t given, and it is focused on developing an understanding of the concepts. Then we go to the 4th question, which requires a much higher level of thinking since I hadn’t yet given them the quadratic formula and they hadn’t worked with factoring, but it is meant as an introduction to the topic (I am curious if this is too far). Since it has no suggested algorithm, it explores relationships and requires significant effort; question 4 pushes it into doing mathematics. Then, question 5 asks the students to make a solution guide, which leads them to explore the connections further, wrapping everything into doing mathematics. (Smith & Stein, 2018) == Why this task fits the standards and the learning goal: == The problem presents a real-life phenomenon that is modeled by a quadratic equation. It begins with a fairly low floor, having students plot a couple of situations, then forcing them to build a quadratic equation for question 3. Part 4 asks students to solve their quadratic equations in a mathematically applicable situation. The problem easily meets the first performance goal, as in the first four questions, the students build and interpret the scenario to create a quadratic equation, then ask the student to solve it in question 4. The second goal is far harder and will mostly fall to monitoring and selecting students who noticed the rate of change. Question 2 will hopefully lead some students to that idea, so I can open that discussion at the end of class. If it hasn’t, I would ask some students questions to help them come across this change. I am hopeful they would consider this rate of change, since, theoretically, it would come from a unit or at least a day of arithmetic. == Motivations == This task can be motivational because it begins with a social and kinesthetic activity: students physically interact, forming groups and shaking hands. This task contextualizes the problem from the start, embedding the problem within an everyday action. This does face a struggle since the connection isn’t particularly meaningful. Middleton and Jansen assert, “Contexts can promote personal investment in mathematics only if they are personally relevant and meaningful for students” (2011). While it isn’t incredibly meaningful, some motivation can be made up because this task is challenging and interesting. Since the problem requires a level of pattern recognition, it can lead to aha moments from students when they see the connection between the handshakes and the quadratic expression. These moments, associated with success at a challenging task, bolsters students’ competence, allowing them to “reap the good feelings associated with competence” ​​(Middleton & Jansen, 2011). = Anticipating Student Solutions = == Solution 1. == === Parts 1 and 2. (This student added up each summation to create the equations, then leveraged the pattern they saw to realize that it was <math display="inline">\frac{n(n-1)}{2}</math>, an important step. It is also helpful that they used the abstract form of the summation. ) === {| class="wikitable" |1 |0 |<math>0=\frac{0n}{2}=\frac{n(n-1)}{2}</math> |- |2 |1 |<math>(n-1)=\frac{1n}{2}=\frac{n(n-1)}{2}</math> |- |3 |3 |<math>(n-1)+(n-2)=\frac{2n}{2}=\frac{n(n-1)}{2}</math> |- |4 |6 |<math>(n-1)+(n-2)+(n-3)=3n-6=\frac{3n}{2}=\frac{n(n-1)}{2}</math> |- |5 |10 |<math>(n-1)+(n-2)+(n-3)+(n-4)=4n-10=\frac{4n}{2}=\frac{n(n-1)}{2}</math> |- |6 |15 |<math>(n-1)+(n-2)+(n-3)+(n-4)+(n-5)=5n-15=\frac{5n}{2}=\frac{n(n-1)}{2}</math> |} '''5 people require 10 handshakes, and the 6th person shakes 5 hands.''' === Part 3. (For part 3 they just plugged the numbers into their equation. They showed that they knew how to plug the inputs into their equations. ) === <math>\frac{30(30-1)}{2}=435 \ handshakes</math> === Part 4. (In part 5 they realized that they needed 2 consecutive numbers to equal 90 so they used guess and check to find the answer. ) === <math>45=\frac{n(n-1)}{2}</math> <math>90=n(n-1)</math> Must be 2 consecutive numbers; guess and check {| class="wikitable" |4 |5 |20 |- |7 |8 |56 |- |12 |13 |156 |- |10 |11 |110 |- |8 |9 |72 |- |9 |10 |90 |} n=10 == Solution 2. == === Part 1. (In part one the student thinks person by person using a table. ) === {| class="wikitable" |E | | | | |X |- |D | | | |X |1 |- |C | | |X |1 |2 |- |B | |X |1 |2 |3 |- |A |X |1 |2 |3 |4 |- | |A |B |C |D |E |} With 5 people, person A shakes 4 new hands, B then shakes 3 new hands, C then shakes 2 new hands, then D shakes 1 new hand, E has already shaken everyone’s hand so they have no new hands to shake so there have been <math>4+3+2+1=10 \ handshakes</math> === Part 2. (The student shows an understanding of how the increase affects the count with the addition of another person.) === {| class="wikitable" |F | | | | | |X |- |E | | | | |X |1 |- |D | | | |X |1 |2 |- |C | | |X |1 |2 |3 |- |B | |X |1 |2 |3 |4 |- |A |X |1 |2 |3 |4 |5 |- | |A |B |C |D |E |F |} With 6 people there are <math>5+4+3+2+1=15 \ handshakes</math> so then there is just 1 more handshake for each of the previous 5 people. '''Therefore the 6th person shakes 5 total hands.''' By plotting these points, we can see a quadratic relationship. Since we know it is a quadratic equation, we can start with 3 people; <math>y=32</math> where everyone shakes two hands and their own Then, since everyone doesn’t shake their own hand we can remove those 3 handshakes Then we are still double counting each handshake so we must divide by 2 to only count each handshake once For part 3 they used a graphical representation of the quadratic expression and a vertical line to find the number of handshakes for each problem. This showed a good understanding of intercepts where they realized that they could use a line to find the y at a certain x. === Part 3. === <math>\frac{30^2-30}2=435 \ handshakes</math> === Part 4. === The student uses a graph and a horizontal line to find when there are 45 handshakes, they showed great understanding of intercepts. They also showed that they understood that the answer could only be positive in the context of this probem. Possible solution 3 that I want to stifle is using the summation in a calculator to easily get the answer because it doesn’t require students to understand the quadratic nature of the problem. == Incorrect Solution == === Part 1 and 2. === The premise behind this solution is that the students working out thought about each person shaking hands and said person 1 shook everyone else’s hand and then person 2 did the same and so on. {| class="wikitable" |1 |0 |<math>0=0n=\frac{n(n-1)}{2}</math> |- |2 |2 |<math>(n-1)+(n-1)=n(n-1)</math> |- |3 |6 |<math display="inline">(n-1)+(n-1)+(n-1)=3(n-1)=n(n-1) \ (each\ person\ shakes\ n-1\ hands)</math> |- |4 |12 |<math>(n-1)+(n-1)+(n-1)+(n-1)=4(n-1)=n(n-1) </math> |- |5 |20 |<math>(n-1)+(n-1)+(n-1)+(n-1)+(n-1)=5(n-1)=n(n-1)</math> |- |6 |30 |<math>(n-1)+(n-1)+(n-1)+(n-1)+(n-1)+(n-1)=5(n-1)=n(n-1)</math> |} The first 5 people shake 20 hands and the sixth adds 10 handshakes. === Part 3. (For part 3 and 4 they just plugged the numbers into their equation. They showed that they knew how to plug the inputs into their equations. ) === 30(30-1)=870 handshakes === Part 5. (In part 5 they knew the factors of 45 and realized that they didn’t have 2 consecutive factors so they declared that it was impossible. ) === 45=n(n-1) Consecutive Factors of 45 {| class="wikitable" |45 |5 |9 |- |45 |3 |15 |- |45 |1 |45 |} Since these are the only factors of 45 and none of them are consecutive, 45 handshakes is impossible. Redirection. To redirect this student I would point them back to when they added the sixth person. I would ask, “when you added your sixth person how many handshakes did you add” They would likely say 10 since that is what they counted. Then I would ask, “Can you count out the handshakes?” If needed I can bring it back into the group and have them act it out and have the struggling student count it out, aloud. Then once they realize it is 5 give them time to think it out and find their mistake on their own. If they are still struggling to bring everything together have them draw out each of the situations (1-5 people) on their paper. = Monitoring and Supporting Student Thinking = === Questions to support progress === Working in groups counting up the handshakes for the first 5 people: '''Assessing''': How are you keeping track of the handshakes to make sure you don’t miss any? '''Advancing''': How might you represent the handshakes? (Leaving it open to various representations, but still advancing them towards a representation.) '''Don’t Say''': Can you make me a graph of these results. (Forces the students towards one representation.) When building a table in solution strategy 1: '''Assessing''': Have you noticed any patterns when you’re counting handshakes? How are you representing the patterns? '''Advancing''': Can you predict how many handshakes would be added for the 7th person? Can you make a formula to predict how many handshakes there would be for any number of people? '''Don’t Say''': "Isn't it just n times n minus 1 divided by 2?" '''If they are struggling to notice the pattern''': Let’s break it down, how many handshakes does each person give? '''Focusing''': If I wanted to find how many handshakes after the 12th person, how would I go about doing that? Then ask hows and whys to try and focus the student understanding. This is based on Herbel-Eisenmann & Breyfogle’s paper, ''Questioning our patterns of Questioning.'' (2005) Once the students have the formula: '''Assessing''': Why do you divide by 2? '''Advancing''': Can you explain the formula within the context of handshakes? When trying to work backwards during part 4 in strategy 1: '''Assessing''': What are you trying to find in this problem? How is this different from the other parts? What information do we know? What strategies are you trying to use in this problem? '''Advancing''': What relationship are you seeing between number of people and number of handshakes? '''Funneling''': How does the formula we found relate to this problem? Can you set up an equation with what you have? '''If students struggle''': It’s ok if this takes a few tries, keep thinking about how the number of handshakes relates to the number of people. Let’s break it down, what information do we know and what are we tring to find? Do we know anything else? From there where do you think we should go? '''What I won’t say''': You need to use the quadratic formula. (we haven’t learned it, and even if we did, it would eliminate the struggle) When using a graphical representation in solution strategy 2 parts 3 and 4: '''Assessing''': What results are you getting from your graph? '''Advancing''': Do the results make sense within the problem? How many results are you getting? Why do you get multiple results? '''If they are struggling to figure out whether to use the positive or negative result''': Within the problem which of these answers makes sense? == What to do for fast finishers?: == If there are fast finishers I will first make sure that their poster is shows their solution strategy well, otherwise they should spend more time on their poster. If the poster is good though I will ask them, "Can you find another solution strategy?", and then once they have I can ask them to make another poster. If multiple groups finish quickly, I can have them send an auditor to another group that is finished who's job is to see if they can follow the other group's poster. I will stress that they are not meant to critique the math, instead they are meant to focus how understandable the poster is. If they can't follow it well they should explain where they got stuck and then the group can work to clarify that portion. There can be multiple rounds of this auditing. == What to do for students just working on the poster?: == The poster is at the end of the task and so unless the student is working on it before they have finished the other problems then I am happy with them working on it until we run out of time. If they seem to focus too much on the artistic stide without showing the math, I can ask, "Can you explain how this poster is showing your solution and explanation?" For all I know they may have a great explanation/plan for how their artistic portion will tie into their solution and I wouldn't want to prevent that. For students working on the poster while working on the other parts I'm fine with that unless they are just doodling before they finish their math. Again I can ask, "Can you explain how this poster is showing your solution and explanation?" and they may have a good explanation. If they don't I can ask them to finish their solution strategy before they work on the poster. = Making Connections and Addressing Learning Goal = The chart above visually represents the mathematical connections between the two student solutions to the handshake problem. Solution 1 uses an Input-Output Table to establish a numerical relationship between the number of people and the corresponding number of handshakes. This table serves as a foundation for identifying the quadratic pattern in the data. By observing the differences in the number of handshakes as the number of people increases, students are able to generalize this pattern into the algebraic expression n(n-1)2​, which I labeled in the chart as the Generalized Quadratic Expression. This process highlights the connection between numerical patterns and their algebraic representations, a key learning goal of this task. Solution 2, on the other hand, employs a Two-Way Table that organizes the new handshakes per person. The table gives the points required for the graphical representation of the quadratic function. This Visualizing Quadratic Function stage demonstrates how the results from the table can be translated into a graphical representation, allowing students to visualize the parabolic curve that represents the handshake relationship. This connection between numerical and graphical representations provides a deeper understanding of the quadratic function and its behavior. The chart also emphasizes the equivalence between the two forms of the quadratic expression, n(n-1)2 and x2-x2​. The Equivalence arrow highlights that different representations can express the same mathematical relationship, further reinforcing the connection between algebraic and geometric solutions. These connections directly align with our learning goals. Firstly, students are expected to build, interpret, and solve a quadratic expression in context. Solution 1 demonstrates the building of the quadratic expression from numerical data. The solution shows how you can use a summation using the input to build a quadratic expression, while Solution 2 focuses on the identifiers in the graphical representation allowing you to realize that it is quadratic. Then Solution 2 builds the expression with the knowledge that it is quadratic. Both solutions ultimately lead to the same algebraic result, showcasing the interconnectivity of these representations. Secondly, students are expected to connect arithmetic equations to quadratic equations by recognizing that while the rate of change of arithmetic equations is constant, the rate of change of the rate of change of quadratic equations is constant. Solution 1 demonstrates this well when you point out that since the output is growing by n-1 each time the nout increases by 1. This shows that the rate of change is increasing by 1 each time. I woul use both of these solutions in group discussions since they both “lead to generalized rules” (Smith & Stein, 2018). == Feedback == A.PAR.6.3: Create and solve quadratic equations in one variable and explain the solution in the framework of applicable phenomena. * I don't think students are likely solve this by creating an equation in this context. I think that this task does a good job of the first part, and this is a precursor of the second. Your learning goal is about comparing with arithmetic which isn't quite what the standard and task is about directly. The fully aligned goal is your first one, but it is more of a performance goal and not so much about what students will learn about. Very thoughtful launch! Solution 1: * Understandings are present but not appropriately specific AND not sufficiently connected to the solution method. == References == Georgia Department of Education. (n.d.). ''GaDOE SuitCASE for Mathematics''. <nowiki>https://case.georgiastandards.org/e9dd7229-3558-4df2-85c6-57b8938f6180/e9dd7229-3558-4df2-85c6-57b8938f6180</nowiki> Herbel-Eisenmann, B. A., & Breyfogle, M. L. (2005). Questioning our patterns of questioning. ''Mathematics Teaching in the Middle School'', ''10''(9), 484-489. Jackson, K., Shahan, E., Gibbons, L., & Cobb, P. (2012). Launching complex tasks. ''Journal of Mathematics Teacher Education'', ''15''(3), 223-242. Middleton, J. A., & Jansen, A. (2011). ''Motivation matters and interest counts: Fostering engagement in mathematics''. National Council of Teachers of Mathematics. ¹ Miller, B. ''The handshake problem''. mrmillermath. (n.d.). <nowiki>https://www.mrmillermath.com/2014/02/06/the-handshake-problem/</nowiki> Smith, M. S., & Stein, M. K. (2018). ''5 practices for orchestrating productive mathematics discussions.'' National Council of Teachers of Mathematics. Warshauer, H. K. (2015). Strategies to support productive struggle. ''Mathematics Teaching in the Middle School'', ''20''(7), 390-393. <nowiki>https://doi.org/10.5951/mathteacmiddscho.20.7.039</nowiki> [[Category:Lesson plans created by prospective teachers]] fsb4f5abci9hcsd327d8zkxarss94wg 0 327235 2807006 2806853 2026-04-29T16:05:43Z Dick Bos 24466 14.1 and 2 2807006 wikitext text/x-wiki This is a resource about the fourth part of the [[Karl Marx/Capital1|first volume of ''Capital'']] by [[Karl Marx]], entitled "'''The Production of Relative Surplus-Value'''". This part holds four chapters: * Chapter 12: The Concept of Relative Surplus-Value * Chapter 13: Co-operation * Chapter 14: The Division of Labour and Manufacture * Chapter 15: Machinery and Large-Scale Industry In the previous part, on the production of absolute surplus value, we have seen the importance of the (length of the) working day in defining (and enlarging) the amount of surplus value. In this chapter Marx delves deeper into the second method of acquiring surplus value: by changing the productivity of the labour process. {| border=0 cellspacing=2 cellpadding=1|| style="width: 100%; background-color: inherit" | style="background-color: #ccc; border: 1px solid #777" | {{center top}}<small>[[Karl Marx/Capital1/Part3|''Capital 1,'' Part 3]] [[Image:Crystal Clear action 1uparrow.png|24px|link=[[Karl Marx/Capital1/Part3]]]]</small>{{center bottom}} | style="background-color: #ccc; border: 1px solid #777" | {{center top}}<small>[[Karl Marx/Capital1/Part5|''Capital 1,'' Part 5]] [[Image:Crystal Clear action 1downarrow.png|24px|link=[[Karl Marx/Capital1/Part5]]]]</small>{{center bottom}} | style="background-color: #bbb; border: 1px solid #666" | {{center top}}<small>[[Karl Marx/Capital1|Karl Marx - ''Capital 1'']] [[Image:Crystal Clear action 2uparrow.png|24px|link=[[Karl Marx/Capital1]]]]</small>{{center bottom}} | style="background-color: #999; border: 1px solid #333" | {{center top}}<small>[[Karl Marx/Critique of Political Economy|Karl Marx's Political Economy]] [[Image:Crystal Clear action 2uparrow.png|24px|link=[[Karl Marx/Critique of Political Economy]]]][[Image:Crystal Clear action 2uparrow.png|24px|link=[[Karl Marx/Critique of Political Economy]]]]</small>{{center bottom}} |} == Chapter 12: The Concept of Relative Surplus-Value == Marx returns to the simple diagram of [[Karl Marx/Capital1/Part3#Chapter 10: The Working Day|chapter 10]]:{{Right|A---------------B------C &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;}} Where AC is the total labour time (the working day), AB is the necessary labour time, and BC is the surplus labour. In chapter 12 he supposes the length of the working-day to be given. Let now the distribution of the working day change as follows: {{Right|A------------B'--B----C &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;}}{{clear}} A smaller part of the working day is spent on necessary labour, and a larger part of surplus labour appears. {{Quote box |quote = I call that surplus-value which is produced by the lengthening of the working day, absolute surplus-value. In contrast to this, I call that surplus-value which arises from the curtailment of the necessary labour-time, and from the corresponding alteration in the respective lengths of the two components of the working day, relative surplus-value. | source = Marx, K. (1867 / 1990). ''Capital 1,'' p. 432}} It is important to note that Marx in this chapter once again refrains from the idea that the capitalist could pay different prices for the same amount of labour: by "pushing the wage of the worker down below the value of his labour-power. (…) Despite the important part which this method plays in practice, we are excluded from considering it here by our assumption that all commodities, including labour-power, are bought and sold at their full value." (p. 431).<ref name=pages>References to page numbers in ''Capital 1'' will be made to the 1990 edition, translated by Ben Fowkes, and published by Penguin Books (which is the same as [https://www.surplusvalue.org.au/Marxism/Capital%20-%20Vol.%201%20Penguin.pdf the 1976 edition of New Left Review]).</ref> As Harvey puts it, this demonstrates "once more Marx's commitment to deconstructing the utopian theses of classical political economy on their own terms."{{sfn|Harvey|2010|p=164}} Producing the same means of subsistence in less hours is impossible without an increase in the productivity of labour, and this can't be done without an alteration in the tools or in the mode of working. "The technical and social conditions of the (labour) process and consequently the mode of production itself must be revolutionized before the productivity of labour can be increased." (p. 432) Shapiro says: "Marx sees relative surplus-value as marking capitalism's substantive shift from being a marginal presence in early modern society to being the dominant one in the modern world. Indeed, capitalism's rrestructuring of production processes can arguably be listed as a key defining feature of modernity."{{sfn|Shapiro|2008|p=97}} As we have seen, the value of a commodity is determined by the socially necessary labour-time congealed in it. When productivity rises, this value diminishes. We have read in [[Karl Marx/Capital1/Part1#Chapter 1: Commodities|chapter 1]]: "In general, the greater the productivity of labour, the less the labour-time required to produce an article, the less the mass of labour crystallized in that article, and the less its value." (p. 131) And in [[Karl Marx/Capital1/Part2#Chapter 6: The Sale and Purchase of Labour-Power|chapter 6]] Marx explained that the "value of labour-power can be resolved into the value of a definite quantity of the means of subsistence. It therefore varies with the value of the means of subsistence, i.e. with the quantity of labour-time required to produce them." (p. 276) In chapter 12 Marx repeats this theory about the value of labour-power<ref name=Petty>Marx draws attention to a passage in [[William Petty]]’s ''Political Anatomy of Ireland'' (1672): The value of the average daily wages of a worker is determined by what the worker needs 'so as to live, labour, and generate.' (p. 430, footnote 1)</ref> and further specifies his statement by writing that, in order "to make the value of labour-power go down, the rise in the productivity of labour must seize upon those branches of industry whose products determine the value of labour-power, and consequently (...) belong to the category of normal means of subsistence." (p. 432)<br /> "But," he remarks, "the value of a commodity is determined not only by the quantity of labour which gives it its final form, but also by the quantity of labour contained in the instruments by which it has been produced. (…) Hence a fall in the value of labour-power is also brought about by an increase in the productivity of labour, and by a corresponding cheapening of commodities in those industries which supply the instruments of labour and the material for labour, i.e. the physical elements of constant capital which are required for producing the means of subsistence. But an increase in the productivity of labour in those branches of industry which supply neither the necessary means of subsistence nor the means by which they are produced leaves the value of labour-power undisturbed." (id.) Marx further notes that each individual capitalist has a motive for cheapening his commodities by continually increasing the productivity of labour. "Capital (…) has an immanent drive, and a constant tendency, towards increasing the productivity of labour, in order to cheapen commodities and, by cheapening commodities, to cheapen the worker himself." (p. 436-437) Here he also finds an answer to a question, already by other political economists, for instance [[Economics/History of Economic Thought/Francois Quesnay|Francois Quesnay]], why capitalist, who want to produce exchange values, continually reduces the exchange value of their commodities. In the following chapters, Marx examines the various historical forms of the development of relative surplus-value. == Chapter 13: Co-operation == "A large number of workers working together, at the same time, in one place (or, if you like, in the same field of labour), in order to produce the same sort of commodity under the command of the same capitalist, constitutes the startingpoint of capitalist production. This is true both historically and conceptually." (p. 440) But apart from the quantitative change, qualitative modifications do take place. First of all, the working together of a large number of workers has the effect that differences between workers are compensated and vanish. In this way "average social labour" becomes a reality.<br /> Then, even "without an alteration in the method of work, the simultaneous employment of a large number of workers produces a revolution in the objective conditions of the labour process." (p. 441) Parts of the means of production are now consumed jointly in the production process. This will lead to lower costs and to a lower price of the commodities produced. The working together of numerous workers, side by side in accordance with a plan, whether in the same process, or in different but connected processes, is called '''co-operation'''. The most simple forms of co-operation will already enlarge productivity, and so do the more complicated forms. Co-operation also allows work to be carried on over a large area. Bringing together large numbers of workers requires a large amount of (variable) capital. But, as more instruments of labour are required, it also requires a large amount of constant capital. Then Marx writes, in his own inimitable style: "Whether the combined working day, in a given case, acquires this increased productivity because it heightens the mechanical force of labour, or extends its sphere of action over a greater space, or contracts the field of production relatively to the scale of production, or at the critical moment sets large masses of labour to work, or excites rivalry between individuals and raises their animal spirits, or impresses on the similar operations carried on by a number of men the stamp of continuity and manysidedness, or performs different operations simultaneously, or economizes the means of production by use in common, or lends to individual labour the character of average social labour - whichever of these is the cause of the increase, the special productive power of the combined working day is, under all circumstances, the social productive power of labour, or the productive power of social labour. This power arises from co-operation itself." (p. 447) Marx further writes that "when the worker co-operates in a planned way with others, he strips off the fetters of his individuality, and develops the capabilities of his species" (p. 447). Harvey remarks that this is one of the instances, "where Marx reverts to some notion of universal species being, which was an important theme in the ''Economic and Philosophical Manuscripts'' of 1844.{{sfn|Harvey|2010|p=173}} {{Blockquote | text = We saw in a former chapter that a certain minimum amount of capital was necessary in order that the number of workers simultaneously employed, and consequently the amount of surplus-value produced, might suffice to liberate the employer himself from manual labour, to convert him from a small master into a capitalist, and thus formally to establish the capital-relation. We now see that a certain minimum amount is a material condition for the conversion of numerous isolated and independent processes into one combined social process.<br />We also saw that, at first, the subjection of labour to capital was only a formal result of the fact that the worker, instead of working for himself, works for, and consequently under, the capitalist. Through the co-operation of numerous wage-labourers, the command of capital develops into a requirement for carrying on the labour process itself, into a real condition of production. That a capitalist should command in the field of production is now as indispensable as that a general should command in the field of battle. | source = p. 448}} This is what according to Harvey must be called the distinction between formal subsumption of labour under capital versus its real subsumption.<ref name=formalvsreal>{{harvnb|Harvey|2010|p=173}} Callinicos criticizes Harvey's interpretation of the distinction between formal and real subsumption {{harvnb|Callinicos|2014|p=200-201}}.</ref> The unification of wage-labourers into a single productive body lies outside the competence of the labourers. "These things are nog their own act, but the act of the capital that brings them together and maintains them in that situation." (p. 449-450) As co-operation extends the capitalist hands over the work of supervision to "a special kind of wage-labourer." (p. 450) Marx mentions managers, foremen and overseers. Co-operation was also a characteristic of works done in ancient Asiatic, Egyptian, and Etruscan societies, and in sporadic forms in the Middle Ages. We must clearly keep in mind that co-operation, contrasted with the process of production carried on by isolated independent workers, is only a specific form of the capitalist process of production. "It is the first change experienced by the actual labour process when subjected to capital." (p. 453) It also forms the real starting-point of capitalist production. "Simple co-operation has always been, and continues to be, the predominant form in those branches of production in which capital operates on a large scale, but the division of labour and machinery play only an insignificant part." (p. 454) This chapter also discusses the concept of "Gesamtarbeit" (social aggregate labour or collective labour) in detail for the first time, although Marx does not yet refer to it by that name.<ref name=gesamt>However, this may also be due to the translation. Compare p. 444, where "Gesamtarbeiter" is translated as "row of men" (rather than "social aggregate labour" or "collective labour"). Incidentally, in MECW vol. 35, p. 332, the translation "collective labour" is used. The MECW ([[:w:en:Marx/Engels Collected Works|Marx/Engels Collected Works]]) is generally regarded as the most authoritative English translation. </ref> The point is that the combined labour of a number of workers working together yields considerably more output than the labour of the same number of workers working individually. Elsewhere in ''Capital 1'', this concept of the "Gesamtarbeit(er)" (social aggregate labour(er) or collective labour(er)) will play an important role. == Chapter 14: The Division of Labour and Manufacture == === Section 1: The Dual Origin of Manufacture === Co-operation in its simple form "remains the fundamental form of the capitalist mode of production, although (…) it continues to appear as one particular form alongside the more developed ones." (p. 454) When co-operation is also based on division of labour, it finds its "classical shape" in manufacture. (p.455) Marx dates this period of the development of capitalism (Shapiro: "The Age of Manufacture"{{sfn|Shapiro|2008|p=105}}, "a gradually increasing move" towards (developed industrial) capitalism) from the middle of the sixteenth century to the last third of the eighteenth century. Marx distinguishes two origins of manufacture: * The "assembling together in one workshop, under the control of a single capitalist, of workers belonging tot various independent handicrafts, through whose hands a given article must pass on its way to completion" (p. 455) * "One capitalist simultaneously employs in one workshop a number of craftsmen who all do the same work, or the same kind of work, such as making paper, type or needles. This is co-operation in its simplest form." (p. 456) Very soon the concentration of workers in one spot will give rise to forms of division of labour. "The commodity, from being the individual product of an independent craftsman<ref name=craftsman>"who does many different things" in the German edition ({{harvnb|Marx|1970|p=358}})</ref>, becomes the social product of a union of craftsmen, each of whom performs one, and only one, of the constituent partial operations." (p. 457) Manufacture grows out of handicrafts in two ways: * "the combination of various independent trades, which lose that independence and become specialized to such an extent that they are reduced to merely supplementary and partial operations in the production of one particular commodity." (…) * "from the co-operation of craftsmen in one particular handicraft; it splits up that handicraft into its various detailed operations, isolating these operations and developing their mutual independence to the point where each becomes the exclusive function of a particular worker." (p. 457) "On the one hand, therefore, manufacture either introduces division of labour into a process of production, or further develops that division ; on the other hand it combines together handicrafts that were formerly separate. But whatever may have been its particular starting-point, its final form is always the same-a productive mechanism whose organs are human beings." (p. 457) But in this kind of co-operation, "handicraft remains the basis." (p. 458) === Section 2: The Specialized Worker and His Tools === If a worker repeatedly does only one aspect of the work process, he will learn to do it faster. "The collective worker ('Gesamtarbeiter' in German), who constitutes the living mechanism of manufacture, is made up solely of such one-sidedly specialized workers. Hence, in comparison with the independent handicraft, more is produced in less time": the productivity of labour increases. Manufacture is also "characterized by the differentiation of the instruments of labour - a differentiation whereby tools of a given sort acquire fixed shapes, adapted to each particular application - and by the specialization of these instruments, which allows full play to each special tool only in the hands of a specific kind of worker." (p. 460) This means that the "work process is broken down into segments that ideally can each be done with a single tool. (…) With a tool for each task, the capitalist no longer needs recalcitrant highly skilled workers who might interfere with the production of surplus-value."{{sfn|Shapiro|2008|p=108}} === Section 3: The Two Fundamental Forms of Manufacture–Heterogeneous and Organic === === Section 4: The Division of Labour in Manufacture, and the Division of Labour in Society === === Section 5: The Capitalist Character of Manufacture === == Chapter 15: Machinery and Large-Scale Industry == This is the most extensive chapter of ''Capital 1'', covering nearly 150 pages. {| border=0 cellspacing=2 cellpadding=1|| style="width: 100%; background-color: inherit" | style="background-color: #ccc; border: 1px solid #777" | {{center top}}<small>[[Karl Marx/Capital1/Part3|''Capital 1,'' Part 3]] [[Image:Crystal Clear action 1uparrow.png|24px|link=[[Karl Marx/Capital1/Part3]]]]</small>{{center bottom}} | style="background-color: #ccc; border: 1px solid #777" | {{center top}}<small>[[Karl Marx/Capital1/Part5|''Capital 1,'' Part 5]] [[Image:Crystal Clear action 1downarrow.png|24px|link=[[Karl Marx/Capital1/Part5]]]]</small>{{center bottom}} | style="background-color: #bbb; border: 1px solid #666" | {{center top}}<small>[[Karl Marx/Capital1|Karl Marx - ''Capital 1'']] [[Image:Crystal Clear action 2uparrow.png|24px|link=[[Karl Marx/Capital1]]]]</small>{{center bottom}} | style="background-color: #999; border: 1px solid #333" | {{center top}}<small>[[Karl Marx/Critique of Political Economy|Karl Marx's Political Economy]] [[Image:Crystal Clear action 2uparrow.png|24px|link=[[Karl Marx/Critique of Political Economy]]]][[Image:Crystal Clear action 2uparrow.png|24px|link=[[Karl Marx/Critique of Political Economy]]]]</small>{{center bottom}} |} == References == {{references}} == Sources == * {{cite book | last = Callinicos | first = Alex | author-link = w:en:Alex Callinicos | date = 2014 | title = Deciphering Capital : Marx's Capital and its destiny | location = London | publisher = Bookmarks Publications | ISBN = 978 1 909026 68 1 | ref = harv }} * {{cite book | last = Harvey | first = David | date = 2010 | title = A Companion to Marx's Capital | location = London / New York | publisher = Verso | ISBN = 978 1 84467 359 9 | ref = harv }} * {{cite book | last = Marx | first = Karl | date = 1970 | title = Das Kapital, Erster Band ''(Marx Engels Werke (MEW) 23)'' | location = Berlin | publisher = Dietz Verlag | ref = harv }} * {{cite web | url = https://www.marxists.org/archive/ruhle/1939/capital.htm | last = Rühle | first = Otto | title = Karl Marx's Capital: An Abridgment | date = 1939 | website = marxists.org | access-date = 2026-01-16 }} * {{cite book | last = Shapiro | first = Stephen | date = 2008 | title = How to Read Marx's Capital | location = London | publisher = Pluto Press | ISBN = 978 0 7453 2562 0 | ref = harv }} [[Category:Karl Marx|Capital]] ai17ae5sluz0w1o86qeolm92pjm2nfh 0 327752 2806988 2806672 2026-04-29T12:44:09Z Solstag 64708 2806988 wikitext text/x-wiki = ''Qualitative-quantitative mixed methods with the Cortext Manager ; methodological background and hands-on practice'' = [[File:Cortext_logo.svg|center|350x350px]] <div style="text-align: center; font-size:1.5em; margin-top:0.5em"> Cortext training course<br/> Institute of Computing, Federal University of Bahia (IC-UFBA)<br/> 27, 28, 29 April 2026 </div> The Cortext Platform is a research infrastructure hosted by the LISIS research unit at Gustave Eiffel University the University (UGE) in France that promotes advanced qualitative-quantitative mixed methods to support researchers on social sciences and humanities. The Cortext training course is an initiative by the LISIS research unit in collaboration with the Institute of Computing (IC) and Institute of Social Sciences (ISC) from Federal University of Bahia (UFBA) in Brazil. The course comprises 2,5 days of theorical and practical training on qualitative and quantitative text data analysis using the Cortext Platform. This course aims at providing methodological and practical skills to analyze scientometric and bibliometric data using Cortext Manager tools and methods, including a quick overview about text analysis, introduction to the Cortext Manager, and data collection from OpenAlex, Scopus and Web os Science (WOS). * '''Dates:''' 27, 28, 29 April 2026 * '''Location:''' Laboratory 140 Institute of Mathematics (IME-UFBA), Campus Ondina (To be confirmed) * '''Target audience:''' UFBA community, especially PhD students, postdoc and researchers * '''Organizing Committee:''' ** Joenio Marques da Costa (UGE, Cortext, IC-UFBA) ** Lionel Villard (UGE, Cortext) ** Christina von Flach (IC-UFBA) ** Claudia Gama (IC-UFBA) ** Leonardo Fernandes Nascimento (ISC-UFBA) * The course will be offered in '''english''' ** Participants will get certification of attending at the end {{center|1=<span style="font-size:1.5em; margin-top:0.5em">[https://grist.numerique.gouv.fr/o/docs/forms/hEZ5aSBf5JEEH2JV2HpnGr/4 Register now]</span>}} == Program == === 27 April 2026, Monday === * '''09h30 - 11h00: Welcome''' ** Introduction: objectives of the training ** Presentation of Cortext platform: context, organization and features * '''11h00 - 12h00: Setting up Cortext Manager''' ** How to access Cortext Manager ** Principles of use ** Setting up the training session project * '''12h00 - 13h00: Lunch break''' * '''13h00 - 17h00: Live demonstration''' ** Presentation of the demonstration subject: worldwide researches on climate change adaptation ** How to design QUERY to delineated a perimeter for the data collection ** Upload and manage the corpus in Cortext Manager ** Explore, how to extract knowledge, how to create lists, how to set up scripts === 28 April 2026, Tuesday === * '''09h30 - 12h00: Learning by doing''' ** Groups constitution based on the type of data and/or the type of subjects of the participants ** Hands on session supported by the Cortext treaners ** Data preprocessing > Upload > Data analysis > Results > Reports ** ''Don't not worry, we can also provide an example: positioning researches driven by researchers located in Brazil in the worldwide landscape on the subject of climate change adaptation'' * '''12h00 - 13h00: Lunch break''' * '''13h00 - 14h30: Insights metrics and algorithms''' ** Distribution and basic statistics ** Metrics of similarities in network ** Network analysis and contingency Matrix * '''14h30 - 17h00: Setting up presentations''' ** Preparing the groups and the participants restitution' === 29 April 2026, Wednesday === * '''09h30 - 11h00: Restitutions''' ** Groups presentations * '''11h00 - 12h00: Final remarks''' ** Recap, feedback, discussions == Online Resources == * Cortext site web: https://www.cortext.net * Access to Cortext Manager: https://managerv2.cortext.net * Cortext project project for use-case demonstration: https://managerv2.cortext.net/project/230470003210 * Repository including all the training materials: https://docs.cortext.net/trainings/digis-sciento-2025 == Organizing Committee == * '''Joenio Marques da Costa''': Joenio is a software engineer at the Cortext Manager project, working to ensure the sustainability of the platform. He is a currently PhD student at PGCOMP UFBA, researching software ecosystem sustainability and evolution. He also nurtures a strong link with the free software communities and Debian project - https://joenio.me/about. * '''Lionel Villard''': Lionel is the head of Cortext Manager and a senior lecturer at ESIEE-Paris, researcher at LISIS laboratory, his research focuses on data mining, scientometrics and data visualizations, dealing with geographical agglomeration and knowledge dynamics - https://www.here-and-there-pics.me/pages/lionel-villard-about. * '''Christina von Flach''': Christina is a senior professor at the Institute of Computing of the Federal University of Bahia since 1990. She holds a PhD degree in Computer Science (2004) and she was the first Director of Graduate Studies (2014-2017) of the Computer Science Program (PGCOMP-UFBA) - the first program to offer both Master’s and PhD degrees in Computer Science in the State of Bahia, Brazil - https://christinaflach.github.io. * '''Claudia Gama''': Claudia is a lecturer at IC-UFBa. She holds a PhD in Interactive Learning Systems from the University of Sussex (2004). She is interest in applied computing in society with emphasis on understanding the impact of technology on people and communities. * '''Leonardo Fernandes Nascimento''': TODO. == Institutional Partners == * [[wikipedia:Gustave Eiffel University|Gustave Eiffel University (UGE)]] * [[wikipedia:Federal University of Bahia|Federal University of Bahia (UFBA)]] [[pt:Cortext/Treinamentos/2026-04-27_UFBA]] 6pyy25b752njdu6op6s01lvzhdf8of2 0 327882 2807035 2806815 2026-04-29T17:10:33Z Mrchristian 281704 2807035 wikitext text/x-wiki DE (EN Below) {{TOCleft}} ==== Linked Open Exhibition ==== ''Materialien und Aufgaben für das Modul „BIM-126-02, SoSe 2026, Worthington/Blümel” für Studierende der Hochschule Hannover. Die Materialien werden gemeinsam mit mehreren Kollegen aus dem [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab] der TIB Hannover erstellt.'' Projekt-GitHub-Repo: https://github.com/NFDI4Culture/linked-open-exhibition ==== Zusammenfassung ==== Der achtteilige Kurs bietet eine Einführung in Linked Open Data (LOD) im Kontext von: # Open Galleries Libraries Archives and Museums (GLAM) und # der Nutzung von Plattformen der Wikimedia Foundation. Die folgenden Plattformen der Wikimedia Foundation werden verwendet: Wikidata, Wikibase, MediaWiki und Wikimedia Commons. AI LLM wird in den folgenden Workflows verwendet: Code Assistant ''Copilot'' und eine Vielzahl von AI LLM-Chat-Diensten für die Dateierstellung und Konfigurationen zur Erstellung von SPARQL-Abfragen, Jinja 2.0-Vorlagen usw. Das „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) kann für unbegrenztes ChatGPT5 genutzt werden: https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat Die verwendeten Methoden sind: Open-Source-Software, Open Science und Rapid Prototyping. ==== Linked Open Exhibition ==== Die Frage, die in diesem Kurs untersucht wird, lautet: Wie kann LOD genutzt werden, um Museumsausstellungen als Linked Open Exhibitions zu verbessern – als Aufzeichnung der Ausstellung, als Katalog der Ausstellungsstücke und für andere wichtige Daten? Als Beispiele '''dienen die Steigerung der Besucherzahlen von Ausstellungen und die Schaffung einer größeren Tiefe des Engagements'''. Der Schwerpunkt liegt auf der Frage, wie LOD-Aufzeichnungen von '''Exponaten in einer Ausstellung''' erstellt werden können. ==== Lernpunkte – in der Reihenfolge ihrer Priorität ==== # '''Wikidata/Wikibase LOD-Konzepte:''' Objekte, Eigenschaften, Werte, Qualifikatoren, Wikibase-Schemas, Klassen, Lexeme, Wissensbasis und Wissensgraphen. # '''Linked Open Data (LOD):''' Semantic Web, 5-Sterne-Bewertung, RDF/Triples, Ontologien, Taxonomien und kontrollierte Vokabulare. # '''Verwendung von LOD-Quellen:''' Identifikatoren, PIDs, Informationsquellen, Medienquellen sowie Import- und Export-Tools. # '''Datenmodellierung:''' Methodiken, Schemaverwendung, Visualisierung und Testen. # '''Daten-Workflow-Tools:''' Git, IDE, KI-Code-Assistent (Copilot), KI-Chat, Verwendung von Wikimedia Foundation-Tools, Datenimport- und -export-Tools, Generierung von PIDs und Hinterlegung in einem wissenschaftlichen Repositorium. # '''Datenpräsentation und Datennutzung:''' Ergebnisse des Wikidata Query Service, MediaWiki-Infoboxen, Verarbeitung von SPARQL-Abfragen durch KI-Chat. # '''Open-Science-Praxis:''' Open-Source-Software, Open Notebook Science, Open Licensing, PIDs, FAIR-Datenprinzipien sowie ethische und bewährte Verfahren bei der Nutzung von KI. ==== Sitzungen ==== Die Sitzungen befassen sich mit der Katalogisierung von Ausstellungen des Sprengel Museums unter Verwendung von LOD und der Erstellung von Visualisierungen und Präsentationen. '''Das Ziel des Lernens ist es, den Umgang''' mit '''LOD''' zu '''erlernen.''' Die Methode besteht darin, ausgehend von einem Kern einer „Ausstellung” „Exponate in einer Ausstellung” hinzuzufügen. Von Anfang an sind es die Studierenden, die die LOD erstellen. Dies beginnt mit minimalen Einträgen der Studierenden, die dann mit Identifikatoren, LOD-Medienquellen, Schemata usw. ergänzt werden. Schließlich wird gezeigt, wie die Daten so präsentiert werden können, dass sie dem „Anwendungsfall” entsprechen: '''die Besucherzahlen der Ausstellungen zu steigern und ein tieferes Engagement zu erreichen'''. Hier kommen Präsentationstechnologien zum Einsatz: MediaWiki-Infoboxen, Ergebnisse des Wikidata Query Service, KI-Chat-SPARQL-Abfragen und andere Funktionen usw. ==== Sitzung 1: Erstellung einer Ausstellung-Zeitleiste – Aufbau, Hinzufügen von Ausstellungen ==== # Erfassen Sie minimale Informationen zu einer Ausstellung in Wikidata als Linked Open Data: Titel, Museum, Datum usw. Beispiel: https://www.wikidata.org/wiki/Q138547468 – Siehe: Tabelle1 : ''Minimale Dateneinträge für eine Ausstellung'' # Zeigen Sie den Ausstellungseintrag in Wikidata an Ergebnisse des Abfragedienstes anzeigen Link (Zeitleiste und Grafik https://w.wiki/J8NJ | https://w.wiki/J8aS ) # Überprüfen Sie die Ausstellungseinträge. # Behandeln Sie Themen, die durch die Erstellung eines LOD-Eintrags aufgeworfen werden: Wikidata-Grundlagen, bewährte Verfahren für Wikidata, Konsultation von Schemata, Bedeutung der Überprüfung und Verwendung von GitHub Issues, Vergleich der verfügbaren Daten – vorher und nachher. ==== Sitzung 2: Ausstellungskatalogisierung – Aufbau, Hinzufügen von Objekten, Künstlern, Katalogen ==== ==== Sitzung 3: Museumsbesuch – Sprengel Museum (noch zu bestätigen) ==== ==== Sitzung 4: Ausstellungskatalogisierung – Massenhinzufügungen: Hinzufügen von Objekten, Künstlern, Katalogen ==== ==== Sitzung 5: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 6: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 7: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 8: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== --- ==== Sitzung 1: Erstellung eines Ausstellungskalenders – Aufbau, Hinzufügen von Ausstellungen ==== Die Übung: Erstellen Sie einen Linked-Open-Data-Datensatz für eine Ausstellung mit Wikidata (Mindestangaben). A. '''Erstellen des Ausstellungseintrags in Wikidata.''' # Anmeldung bei Wikidata: https://www.wikidata.org/ # Halten Sie eine Quelle bereit, um Daten einzugeben, z. B. #* https://www.sprengel-museum.de/ausstellungen/archiv #* https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised #* https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&currentPosition=1 # Überprüfen Sie, ob es bereits einen Eintrag für die Ausstellung auf Wikidata gibt. Verwenden Sie dazu die Suchfunktion. # Erstellen Sie einen Eintrag oder bearbeiten Sie einen bestehenden Eintrag. #* Hinweis: Überprüfen Sie, welche Sprache Sie verwenden. Wir werden Einträge in Deutsch und Englisch hinzufügen (beginnend mit Deutsch). # Erstellen Sie die folgenden Dateneinträge in Wikidata, siehe: Tabelle 1: ''Minimale Dateneinträge für eine Ausstellung.'' # Überprüfen Sie die Wikidata-Einträge zur Ausstellung. Die Überprüfung erfolgt anhand von drei Fragen. Fügen Sie bei Bedarf Kommentare hinzu, Korrekturen können vorgenommen werden. Ergebnisse und Anmerkungen können auf der Diskussionsseite des Eintrags hinzugefügt werden, z. B. #* Alle Einträge vorhanden [ ] #* Alle Einträge sind korrekt [ ] #* Einträge sind in Deutsch und Englisch – im Rahmen des Zumutbaren [ ] ''Tabelle2 : Mindestdaten für einen Ausstellungseintrag'' {| class="wikitable" | colspan="7" |'''Felder, die zur Erstellung eines Ausstellungseintrags verwendet werden. Siehe Beispiel: https://www.wikidata.org/wiki/Q138547468''' |- |A |Beschriftung | colspan="5" |Hinweis: Kurz halten. Titel der Ausstellung verwenden |- |B |Beschreibung | colspan="5" |Hinweis: Zur Unterscheidung von anderen Einträgen verwenden. Folgen Sie diesem Beispiel: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Eigentum (P) und Objekt (Q)''' |'''URI''' |'''DE''' | |'''Hinzufügen''' |'''Anmerkung''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |Instanz von |Q464980 |Element hinzufügen |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Ausstellung | |(oben verwendet) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Titel |Titel |Klartext |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Standort |Sprengel Museum Hannover Q510144 |Artikel hinzufügen |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Startzeit |Datum |JJJJ-MM-TT |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |Endzeit |Datum |JJJJ-MM-TT |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Kurator |Person |Element hinzufügen (falls nicht vorhanden, muss erstellt werden/kann derzeit weggelassen werden) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Teilnehmer |Person (der Künstler) |Element hinzufügen (falls nicht vorhanden, muss erstellt werden/kann derzeit weggelassen werden) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |Offizielle Website |Offizielle Website |URL |URL |} Ende von Sitzung 1. ==== Hausaufgabenübungen ==== 1. Vervollständigen Sie Ihre zugewiesene Ausstellung. Stellen Sie sicher, dass alle Felder aus Tabelle 1 ausgefüllt sind. Wenn etwas nicht hinzugefügt werden kann, haben Sie zwei Möglichkeiten: A. Machen Sie eine Notiz in der Tabelle zur Ausstellungszuweisung oder B. Senden Sie eine E-Mail an [mailto:Simon.worththington@tib.eu simon.worththington@tib.eu] , damit ich Ihnen bei der Lösung Ihres Problems helfen kann. '''Hinweis: Wenn Sie während des Unterrichts keinen Ausstellungseintrag erstellt haben, stellen Sie sicher, dass dieser vor der nächsten Unterrichtsstunde fertiggestellt ist.''' 2. Erstellen Sie ein GitHub-Konto und fügen Sie Ihren GitHub-Namen neben Ihrem Namen in der Spalte „GitHub-Name” in der Tabelle zur Zuweisung der Ausstellungen hinzu. 3. Überprüfen Sie die Ausstellungseinträge Ihrer Klassenkameraden. Ihnen wurde allen ein Eintrag zur Überprüfung zugewiesen, siehe Tabelle zur Zuweisung der Ausstellungen. Ihr Name steht in Spalte G. Diese erste Überprüfung umfasst drei Fragen – kreuzen Sie die Kästchen an, um anzuzeigen, ob jeder Punkt ausgefüllt wurde, und fügen Sie entweder Kommentare hinzu oder korrigieren Sie den Wikidata-Ausstellungseintrag. '''Hinweis: Wenn der Ihnen zugewiesene Ausstellungseintrag nicht von Ihrem Klassenkameraden erstellt wurde, kontaktieren Sie ihn bitte und bitten Sie ihn, den Eintrag zu vervollständigen.''' Die Fragen lauten: 1. Sind alle erforderlichen Felder vorhanden? 2. Sind alle Felder korrekt ausgefüllt? 3. Gibt es einen deutschen und einen englischen Eintrag? --- === Sitzung 2: Ausstellungskatalogisierung – Aufbau, Hinzufügen von Objekten, Künstlern, Katalogen === ==== Die Sitzung umfasst fünf Übungen: ==== # Ausstellungsaktualisierung # Künstler # Ausstellungskatalog # AI LLM SPARQL-Experimente # <s>Kunstwerk</s> ==== Die Übungen umfassen die folgenden Konzepte: ==== ==== Übungen ==== ==== 1. Aktualisierung der Ausstellung ==== * Hausaufgabenüberprüfung: Füllen Sie alle Felder für eine Ausstellung aus. Überprüfen Sie die Ihnen zugewiesene Ausstellung, indem Sie die folgenden drei Fragen beantworten: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch</blockquote> * Für das Label. Wandeln Sie Wörter in Großbuchstaben in Satzschrift um. Verwenden Sie: https://convertcase.net/title-case-converter/ | Ändern Sie z. B. ADRIAN SAUER: TRUTH TABLESPECTRUM INTERNATIONALER PREIS FÜR FOTOGRAFIE DER STIFTUNG NIEDERSACHSEN in Adrian Sauer: Truth Tablespectrum Internationaler Preis Für Fotografie Der Stiftung Niedersachsen. * Fügen Sie die englischen Versionen hinzu. Verwenden Sie DeepL zum Übersetzen: https://www.deepl.com/en/translator ** Titel: Fügen Sie den englischen Titel hinzu * Fügen Sie Folgendes hinzu. Ändern Sie P710 Teilnehmer (Participant) in P921 zentrales Thema artists name. ** Qualifier zum zentralen Thema, um anzugeben, dass die Person Kunstwerke beisteuert. * Verwenden Sie: Qualifier P170 creator und fügen Sie artist Q483501 hinzu (geben Sie „Künstler” ein, es wird automatisch vervollständigt) * Referenz: Gemeinsame Normdatei (GND) ID für eine Person, z. B. Gabriela Jolowicz https://d-nb.info/gnd/134184963 | Suchen Sie den Namen der Person und kopieren Sie den letzten Teil der Nummer 134184963 * Diskussionsseite: Fügen Sie die Überprüfungsfragen für Ihren Wikidata-Eintrag hinzu: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch?</blockquote>Beachten Sie die nützlichen Links, die Ihnen mehr über verbundene Linked Open Data verraten! Hinweis: SPARQL-Abfrage zur Anzeige des Datenmodells. Eigenschaften und Werte. Ergebnisse: https://w.wiki/JMLX Erstellt mit Gemini AI: https://gemini.google.com/share/c43f34a67f67 ==== Konzepte ==== * Wikidata-Teile – siehe Informationen und Diagramm: ** https://www.wikidata.org/wiki/Wikidata:Introduction/de ** https://www.wikidata.org/wiki/Wikidata:Introduction#/media/File:Datamodel_in_Wikidata.svg * Anwendung eines Überprüfungsprozesses mithilfe von Diskussionsseiten * Hinzufügen von Referenzen * Verwendung einer LOD-Quelle – Ein Normdatensatz Gemeinsame Normdatei (GND) ID <nowiki>https://portal.dnb.de/opac.htm</nowiki> * SPARQL-Abfrage --- ==== 2. Künstler ==== Das Ziel hierbei ist es, sicherzustellen, dass alle Künstler in die Ausstellungsliste aufgenommen wurden, und anschließend die bestehenden Künstlereinträge zu überprüfen. Später wird eine SPARQL-Abfrage durchgeführt, um Aussagen über alle Künstler in unserem Datensatz zu vergleichen. Bevor Sie die Künstereinträge überprüfen, stellen Sie sicher, dass alle Künstler im Ausstellungseintrag aufgeführt sind, mit dem Qualifikationsmerkmal „Künstler” und einem Verweis auf ihren GND-Datensatz. ==== Wichtige Aussagen ==== {| class="wikitable" |Concept |CIDOC CRM (Full) |Linked Art (Selection) |Wikidata Equivalent |Note |- |Entity |E21 Person |Person |Q5 (human) |The base instance. |- |Label/Name |P1 is identified by → E33_E41 |identified_by (Name) | |Linked Art flattens this into a simple list of names. |- | | | |P735 Given name | |- | | | |P734 Family name | |- |Profession |P2 has type → E55 Type |classified_as |P106 (occupation) |Map to AAT 300025103 (artist). |- |Birth |P98i was born → E67 Birth |born (Birth) |P569 (date of birth) |CRM treats birth as an event; Wikidata as a property. |- |Death |P100i died in → E69 Death |died (Death) |P570 (date of death) |If the artist is still living, this is omitted. |- |Nationality |P107i member of → E74 Group |classified_as (Type) |P27 (citizenship) |Linked Art often models nationality as a Type. |- |Reference |P1 identifies ← E42 Identifier |identified_by (Identifier) |QID (The URI itself) |Used to link to external authorities (ULAN, VIAF). |- |Commons category |? |? |P373 search name |<nowiki>https://commons.wikimedia.org/</nowiki> |} Aus Google Gemini: https://gemini.google.com/share/578cc1b886d0 --- ==== Schemas und Communities benötigen Beratung. ==== '''Aus Wikimedia:''' WikiProject Visual Arts: https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Visual_arts Wikiproject Exhibitions: https://www.wikidata.org/wiki/Wikidata:WikiProject_Exhibitions '''Halbformell''' Generisches Wikibase-Modell für Kulturdaten: https://kgi4nfdi.github.io/Guidelines/guide/wikibase/data_modelling_import/ '''Formell:''' CIDOC Conceptual Reference Model (CRM) – https://cidoc-crm.org/ Linked Art (basierend auf CIDOC) https://linked.art/model/actor/ ==== Konzepte ==== * Datenmodellierung * Schemas * Anwendungsfall * Bottom-up-Design * Identifikatoren --- ==== 3. Ausstellungskatalog ==== Suchen Sie an beiden Orten nach Informationen zum Katalog Ihrer zugewiesenen Ausstellung. Sprengel Museum Publikationskatalog – https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised DND (Beispiel) Sie können nach dem Namen der Ausstellung oder dem Sprengel Museum suchen – https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true Hinweis: Notieren Sie sich alle Links, die Sie in der Tabelle mit den Ausstellungslisten finden. ===== Erstellen Sie einen Wikidata-Eintrag für den Katalog. ===== Hinweis: Suchen Sie zunächst nach der Veröffentlichung, bevor Sie einen Wikidata-Eintrag erstellen. Verwenden Sie den Titel, die ISBN und die GND. Ein Beispiel für eine Veröffentlichung aus DNB und Sprengel Shop. * https://portal.dnb.de/opac/showFullRecord?currentResultId=Gabriela+and+Jolowicz%26any&currentPosition=0 * https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised ===== Geben Sie diese Angaben ein ===== Hinweis: Denken Sie an die Bezeichnung und Beschreibung {| class="wikitable" |Property |Label |Description/Example |- |P31 |instance of |catalogue (Q2352616) |- |P1476 |title |The official title of the catalogue (e.g., Vermeer and the Masters of Genre Painting) |- |P50 |author |The main curator or art historian (item link) |- |P123 |publisher |The museum or publishing house (e.g., Louvre Museum) |- |P577 |publication date |Year of release (e.g., 2024) |- |P212 |ISBN-13 |The 13-digit standard book identifier |- | |GND |ID |- |P973 |described at URL |A link to the catalogue's page on the museum’s website |} Google Gemini https://gemini.google.com/share/9a21f5522192 Beispiel für eine Eingabe: https://www.wikidata.org/wiki/Q138646145 ==== Verlinken Sie den Datensatz zurück zur Ausstellung. ==== P972 > Titel ==== Konzepte ==== * Datenmodellierung * Identifikator * Daten als CC Zero / Urheberrecht der Daten --- ==== 4. AI LLM SPARQL-Experimente ==== Wikidata verfügt über eine SPARQL-Schnittstelle, über die die LOD in Wikidata durchsucht (abgefragt) und auf verschiedene Arten, in verschiedenen Formaten und Visualisierungen ausgegeben werden kann. Außerdem kann sie im Web gespeichert werden. Wir werden den AI LLM-Chat verwenden, um SPARQL-Abfragen zu generieren. Später werden wir die Grundlagen des Schreibens einer SPARQL-Abfrage lernen. Aber zunächst wollen wir sehen, wie sie generiert werden, welche Optionen es gibt und wie sie kreativ eingesetzt werden können. Die Verwendung von Chat-Diensten oder Code-Assistenten kann eine wertvolle Möglichkeit sein, um neue Technologien kennenzulernen. {| class="wikitable" |Service |Best For |Standout Feature |Key Model(s) |- |'''ChatGPT''' |General Use & Tasks |Deep Research & Agent Mode |GPT-5.4, GPT-5 |- |'''Claude''' |Coding & Writing |Artifacts (interactive workspace) |Claude 4.5, 4.6 |- |'''Google Gemini''' |Google Ecosystem |Nano Banana (native image/video) |Gemini 3.1 Pro |- |Perplexity |Real-time Research |Native Citations & Search Labs |Sonar, GPT-5, Claude |- |MS Copilot |Office Productivity |Copilot Vision & 365 Integration |GPT-5.2, Prometheus |- |DeepSeek |Logical Reasoning |High-tier performance at low cost |DeepSeek-V3, R1 |- |Grok |Real-time Social Info |Unfiltered X (Twitter) integration |Grok 4.1 |- |'''Meta AI''' |Social Media |Seamless integration in WhatsApp/IG |Llama 4 (Scout) |- |Poe |Model Testing |Access multiple LLMs in one app |Multi-model aggregator |- |Mistral (Le Chat) |Privacy & Developers |European-hosted, GDPR-focused |Mistral Large 3 |} Einige davon können auch über KISSKI genutzt werden. Das „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) kann für unbegrenztes ChatGPT5 genutzt werden: https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat ==== Die Übung ==== Die Gruppe wird in mehrere Zoom-Breakout-Gruppen aufgeteilt und verbringt dann 20 Minuten damit, SPARQL-Abfragen und andere kreative Anwendungen zu generieren. Fügen Sie die Ergebnisse hier ein: https://tib.cloud/apps/files/files/8251374?dir=/NFDI4Culture/HsH/BIM26/bim26-shared&editing=false&openfile=true Jedem Raum wird eine Chat-Engine zugewiesen. Es gibt maximal vier Gruppen. · Gruppe Nr. 1: ChatGPT · Gruppe Nr. 2: Claude · Gruppe Nr. 3: Google Gemini · Gruppe Nr. 4: Meta AI ==== Beispielübung ==== Chatbots können eine SPARQL-Abfrage oder eine Wikidata-Adresse lesen. z. B. * Artikel https://www.wikidata.org/wiki/Q138547468 * Abfragegrafik https://w.wiki/JPNc * Abfragetidsachse https://w.wiki/JPPN * Artikel Sprengel Museum https://www.wikidata.org/wiki/Q510144 Anschließend kann der Chatbot angewiesen werden, auf Grundlage der bereitgestellten Informationen bestimmte Aktionen auszuführen. Sie sollten den Chatbot bitten, Wikidata-SPARQL-Abfragen zu generieren, und diese Abfragen dann in die SPARQL-Abfrageoberfläche einfügen. https://query.wikidata.org/ Verwenden Sie diese Beispiele und entwickeln Sie Ihre eigenen: # Dashboard erstellen (Anzahl der Dinge) # Inventar erstellen (Tabelle) # Graphdatenmodell erstellen Einige SPARQL-Abfragen · Karte der Geburtsorte von Künstlern – https://w.wiki/JPT3 · Liste der Ausstellungen – https://w.wiki/JPR3 · Als Darstellung der Ausstellungen – https://w.wiki/J8aS ==== Hausaufgabe: Sitzung 2 ==== Erstellen Sie ein Bottom-up-Datenmodell eines Kunstwerks in einer Ausstellung. Fügen Sie nur die minimal erforderlichen Informationen hinzu. Das Ergebnis sollte eine Tabelle sein, wie sie für Ausstellung, Künstler und Katalog dargestellt wird. Die Tabelle sollte Eigenschaften und Attribute enthalten. Sie sollten die oben genannten Schemata zu Rate ziehen. Sie können KI verwenden, aber geben Sie die KI an und verlinken Sie sie mit Ihrer Frage. Wenn Sie KI verwenden, überprüfen Sie die Ergebnisse und machen Sie sich Notizen darüber, was Sie geändert haben. Hinweis: Überlegen Sie, wie die Teile miteinander in Beziehung stehen, was Sie hinzufügen müssen und was bereits in Wikidata vorhanden ist. Reichen Sie Ihre Ergebnisse als Tabelle oder Spreadsheet ein. --- ==== Sitzung 3: Museumsbesuch – Sprengel Museum ==== 19 März 2026 ENDE ==== Sitzung Nr. 4: Schemata und Prototyping (Abschlussprojekt) ==== ===== Zusammenfassung und Überblick ===== Erledigt * Erstellen von Ausstellungs-Einträgen in Wikidata * Befüllen unserer Datenmodelle für „Künstler“ und „Katalog“ * Erkundung des Museums und seiner Aktivitäten, um den Prototyp zu steuern Zu erledigen * Entscheidung über die Ideen für den Prototyp * Datenmodell für Objekte in einer Ausstellung (Kunstwerk und Ausstellung) * Erstellen eines Datenmodells zum Projektende, das von Museen genutzt werden kann und den Branchenstandards – CIDOC und Wikidata – entspricht. ===== Was haben wir über die „Geschichte des Museums“ gelernt? ===== TBC ===== Schemas ===== Eine Gelegenheit, sich mit der Struktur von Linked Open Data anhand gemeinsamer Vereinbarungen zu Arbeitspraktiken vertraut zu machen. Im Laufe des Kurses wird ein Datenmodell entwickelt und fertiggestellt, um „Objekte in einer Ausstellung“ zu beschreiben. Das Datenmodell wird zur Konsultation und zum Testen durch die Community veröffentlicht. ===== Schemas und Schlüsselkonzepte ===== Tabelle: https://tib.cloud/s/ZKNAAo3B8ATXsAP * Schema * Terminologiedienst * Kontrolliertes Vokabular * Taxonomie * Ontologie * Wissensgraph Tabelle X: Link: https://tib.cloud/s/ZKNAAo3B8ATXsAP In Linked Open Data (LOD) verwendete Terminologie DE {| class="wikitable" ! Konzept ! Wikidata-Link (Konzept) ! Hauptschwerpunkt ! Analogie ! Beispielressource ! URL ! Anwendungsbeispiel ! URL |- | Schema | Q1397073 | Datenstruktur | Die Vorlage. Konzeptionelles Schema / Datenmodell | Schema.org | [https://schema.org/] | VisualArtwork | [https://schema.org/VisualArtwork] |- | | | | | | | Smithsonian American Art Museum (SAAM) „In der Sierra Nevada, Kalifornien“ | [https://www.wikidata.org/wiki/Q20475372] |- | Terminologiedienst | Q22692845 | Verbreitung | Eine Bibliothek mit Vokabularen, Schemata, Ontologien usw. | TIB-Terminologiedienst | [https://terminology.tib.eu/ts/] | NFDI4CULTURE | [https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE] |- | Kontrolliertes Vokabular | Q1469824 | Konsistenz | Das Wörterbuch | Integrierte Normdatei / die Gemeinsame Normdatei (GND) | [https://portal.dnb.de/opac/showShortList] | Personen: Dürer, Albrecht | [https://d-nb.info/gnd/117751669] |- | Taxonomie | Q8269924 | Hierarchie | Sortierung nach Typ (allgemeine Klassifizierung) | Getty Art & Architecture Thesaurus (AAT) | [https://www.getty.edu/research/tools/vocabularies/aat/] | Deutscher Surrealist Max Ernst (verwendete Maltechniken) |[https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20und%20Grattage,in%20seinen%20Zeichnungen%20von%201925]. |- | | | | | Iconclass | [https://iconclass.org/] | Max Ernsts „Die Jungfrau, die das Christkind versohlt“ (Parady) | [https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926] |- | Ontologie | Q324254 | Semantik: Bedeutung & Logik (Informationswissenschaft) | Das Regelwerk oder der Stilführer | CIDOC (Comité International pour la DOCumentation / Internationales Komitee für Dokumentation) | [https://cidoc-crm.org/] | Sloane Lab Knowledge Base – Zusammenführung von 3 Sammlungen | [https://knowledgebase.sloanelab.org/resource/Start] |- | Wissensgraph | Q33002955 | Netzwerk von Dingen und Beziehungen | Eine Navigationskarte | Verzeichnis antiker Kunstwerke und architektonischer Bauwerke, die in der Renaissance bekannt waren | [https://www.census.de/] | Artemis-Suche | [https://database.census.de/#/detail/10013099] |- | | | | | Forschungsbereich | [https://researchspace.org/] | Hokusai: Das große Bilderbuch von allem |[https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start] |} EN {| class="wikitable" |- ! **Concept** ! **Wikidata link (Concept)** ! **Primary Focus** ! **Analogy** ! **Example resource** ! **URL** ! **Example use** ! **URL** |- | Schema | Q1397073 | Data Structure | The Template. Conceptual schema / data model | Schema.org | https://schema.org/ | VisualArtwork | https://schema.org/VisualArtwork |- | | | | | | | Smithsonian American Art Museum (SAAM) "Among the Sierra Nevada, California" | https://www.wikidata.org/wiki/Q20475372 |- | Terminology Service | Q22692845 | Distribution | A Library of Vocabularies, Schemas, Ontologies, etc | TIB Terminology Service | https://terminology.tib.eu/ts/ | NFDI4CULTURE | https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE |- | Controlled Vocabulary | Q1469824 | Consistency | The Dictionary | Integrated Authority File / die Gemeinsame Normdatei (GND) | https://portal.dnb.de/opac/showShortList | Persons: Dürer, Albrecht | https://d-nb.info/gnd/117751669 |- | Taxonomy | Q8269924 | Hierarchy | Sorting things by type (general classification) | Getty Art & Architecture Thesaurus (AAT) | https://www.getty.edu/research/tools/vocabularies/aat/ | German Surrealist Max Ernst (painting techniques used) | https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20and%20Grattage,in%20his%20drawings%20in%201925. |- | | | | | Iconclass | https://iconclass.org/ | Max Ernst’s "The Virgin Spanking the Christ Child" (Parady) | https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926 |- | Ontology | Q324254 | Semantics: Meaning & logic (information science) | The Rulebook or Writing Style Guide | CIDOC (Comité International pour la DOCumentation / International Committee for Documentation) | https://cidoc-crm.org/ | Sloane Lab Knowledge Base - unifying 3 collections | https://knowledgebase.sloanelab.org/resource/Start |- | Knowledge Graph | Q33002955 | Network of things and relations | A Navigational Map | Census of Antique Works of Art and Architecture Known in the Renaissance | https://www.census.de/ | Artemis search | https://database.census.de/#/detail/10013099 |- | | | | | Research Space | https://researchspace.org/ | Hokusai: The Great Picture Book of Everything | https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start |} ===== Schema-Übung ===== Zu bearbeitende Tabelle: https://tib.cloud/s/PicTdwCEqCQ6pBp (Passwort: bim2026) Wir werden uns mit folgenden Themen befassen: Ausstellung, Künstler und Katalog. '''''Geben Sie die gefundenen URLs ein. Fügen Sie bei Bedarf neue Zeilen, Spalten und Kommentare hinzu. Führen Sie sowohl manuelle als auch KI-Suchen durch, um die Ergebnisse zu vergleichen.''''' ===== Übung Nr. 1: Trage Links zu passenden Elementen aus den folgenden Quellen in die Tabelle ein: ===== * Wikidata:WikiProject Exhibitions/Properties * Generisches Wikibase-Modell für Kulturdaten – Wikibase4Research NFDI4Culture * CIDOC CRM (vollständig) * Terminologiedienst (NFDII4Culture) * Wikidata ===== Übung Nr. 2: Verwenden Sie KI-LLM, um passende Elemente zu finden ===== * <nowiki>https://gemini.google.com/</nowiki> ==== Prototyping ==== Entweder in dieser oder in der nächsten Sitzung wird die Gruppe in Teams aufgeteilt. ===== Schema ===== # Entwicklung eines Datenmodells: „Objekte in einer Ausstellung“ ===== Teile der Quarto-Publikation ===== # Ein Katalog einer Ausstellung des Sprengel Museums # Ein Katalog aller Ausstellungen und Ausstellungskataloge # Katalog der Ausstellungsbeiträge # Ein Glossar mit Begriffen – Personen und benannte Entitäten – aus Wikidata --- ===== Einführung in Quarto und Einfügen eines Ausstellungsbeitrags ===== Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace, Copilot: Agentic Coding) '''Voraussetzungen''' # Ein Laptop oder Computer, auf dem Sie VScode installieren können # Sie benötigen 2FA auf Ihrem Mobilgerät # Erstellen Sie ein GitHub-Konto # Installiere VScode # Verbinden Sie Ihr GitHub-Konto mit VScode # Erstellen Sie ein GitHub-Repository '''Klonen:''' https://github.com/mrchristian/prototype '''Modell: Auto''' '''So wurde das Repo eingerichtet. Agent-Eingabeaufforderungen:'''<blockquote>Ich möchte ein Quarto-Website-Projekt ausführen, bitte richte die Grundlagen ein. Das Projekt wird auf GitHub Pages veröffentlicht. Lege das Ausgabeverzeichnis auf „docs“ fest.</blockquote>Erstellen Sie eine Seite für das Quarto-Projekt, die die für diesen Wikidata-Eintrag verwendeten Daten abruft und als professionelle Webseite rendert <Fügen Sie hier Ihre Ausstellung ein – oder verwenden Sie diese> https://www.wikidata.org/wiki/Q138547468 Der Ansatz sollte eine SPARQL-Abfrage für die Daten erstellen und diese dann mithilfe eines Jupyter-Notebooks als HTML rendern. Alle Einträge: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Aufgaben ===== * Ausstellung ändern * Notebook ausführen * Quarto ausführen und Vorschau anzeigen * Auf Ihren GitHub Pages veröffentlichen ===== Schritt für Schritt ===== '''Teil 1: Arbeitsumgebung''' '''''HINWEIS: Sollten bei der lokalen Ausführung Probleme auftreten, nutzen Sie bitte die Online-Option von Codespace.''''' # Erstelle ein GitHub-Konto – https://github.com/ # Richte die Zwei-Faktor-Authentifizierung (2FA) ein – in der Regel auf dem Handy (Google Authenticator) # Installiere VSCode – https://code.visualstudio.com/download # Installiere GitHub Desktop – https://desktop.github.com/download/ # Füge dein GitHub-Konto hinzu, wenn du dazu aufgefordert wirst, und verwende die 2FA Schritt 2: Der Prototyp # Forken Sie das Repository: https://github.com/mrchristian/prototype # Wenn Sie lokal arbeiten, fahren Sie fort – wenn Sie Codespace verwenden, starten Sie Codespace (siehe unten und fahren Sie dann fort) # Testen Sie Quarto im Terminal: ## quarto check ## quarto render ## quarto preview (Strg+C – zum Beenden) # Falls es nicht funktioniert, führen Sie Quarto über den Agent aus # Ändern Sie die Wikidata-Ausstellung im Notebook # Notebook ausführen # quarto render und quarto preview ausführen # Alles speichern # Git: Nachricht, Commit und Push # Auf GitHub.com dein Repository ## Seiten aktivieren: GitHub Actions ## Code: Über das Zahnrad – Klicke auf „Meine GitHub Pages verwenden“ ## Registerkarte „Actions“: Quarto-Projekt veröffentlichen # ENDE – Wiederholen :-) ===== Codespace-Option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace ist eine virtuelle Maschine, die über GitHub gestartet werden kann. Das Repository enthält eine Dev-Container-Konfiguration, sodass du vollständig im Browser arbeiten kannst, ohne etwas lokal installieren zu müssen. 1.    Klicke auf der Repository-Seite auf GitHub auf „Code“ → „Codespaces“ → „Codespace erstellen“ auf der Hauptseite. 2.    Warte, bis der Container erstellt ist – Python-Pakete aus der Datei „requirements.txt“ werden automatisch installiert – dies dauert etwa 5 Minuten. 3.    Sobald alles installiert ist, kann der Codespace jederzeit genutzt werden. Er fährt automatisch herunter, wenn er nicht genutzt wird, und kann jederzeit neu gestartet werden. 4.    In Codespace geleistete Arbeit muss zurück ins Repository gepusht werden. 5.    Wenn Codespace 28 Tage lang nicht genutzt wird, wird der Codespace gelöscht. --- ===== Hausaufgabe – Sitzung Nr. 4 ===== * Hol alle Bücher aus der HsH-Bibliothek, die Ausstellungskataloge des Sprengel Museums sind. Bring sie zur nächsten Vorlesung mit * Erstelle einen Ausstellungs-Eintrag, falls noch nicht geschehen * Arbeite mit VSCode und dem Agent und experimentiere ==== Sitzung 5: Prototypenerstellung: Running Quarto Prototype, Federation, Prototype Teams ==== * Running Quarto Prototype - wie oben - https://github.com/mrchristian/prototype * DNB data download https://github.com/NFDI4Culture/linked-open-exhibition * Data Federation - WB4R ===== Links ===== https://wikibase.wbworkshop.tibwiki.io/wiki/Main_Page Glossar - https://nfdi4culture.github.io/linked-open-exhibition/Documentation/glossary ===== DNB Suche ===== https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books Sprengel Museum, 602 Artikel '''Über die Dienstleistungen von DNB''' https://www.dnb.de/librarylab https://deutsche-nationalbibliothek.github.io/jupyterlite/lab/ '''AUCH''' https://wiki.dnb.de/spaces/LINKEDDATASERVICE/pages/449878933/DNB+SPARQL+Service+BETA ===== Prototype Teams ===== * DNB-Daten * Katalog durchsuchen * Ausstellungsbeiträge * Vollständiges Datenmodell (alle) ENDE --- == EN == ''Materials and Tasks for the module "BIM-126-02, SoSe 2026, Worthington/Blümel" for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.'' Project GitHub repo: https://github.com/NFDI4Culture/linked-open-exhibition ==== Summary ==== The eight session course covers an introduction to Linked Open Data (LOD) in the context of : # Open Galleries Libraries Archives and Museums (GLAM), and; # The use of Wikimedia Foundation platforms. The Wikimedia Foundation platforms that will be used are: Wikidata; Wikibase, MediaWiki, and Wikimedia Commons. AI LLM will be used in the workflows: Code assistant ''copilot'', and a variety of AI LLM chat services for file generation and configurations to create SPARQL queries, Jinja 2.0 templates, etc. „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat The Methodologies employed are: Open-source software, Open Science, and rapid prototyping. ==== Linked Open Exhibition ==== The question being explored for the class is how can LOD be uséd to benefit museum exhibitions as Linked Open Exhibitions – a record of the exhibition, a catalogues of items in an exhibition, and other important data? As examples '''to gain exhibitions increased visitors numbers and create greater depth of engagement'''. With a focus of the question on how to make LOD records of '''items in an exhibition'''. ==== Learning points – In order of priority ==== # '''Wikidata/Wikibase LOD concepts:''' Items, Properties, Values, Qualifiers, Wikibase schemas, Classes, Lexemes, Knowledge Base, and Knowledge Graphs. # '''Linked Open Data (LOD):''' Semantic web, 5 star, RDF/Triples, Ontologies, Taxonomies, and controlled vocabularies. # '''Using LOD source:''' Identifiers, PIDs, information sources, media sources, and import and export tooling. # '''Data modelling:''' Methodologies, schema use, visualisation, and testing. # '''Data workflow tools:''' Git, IDE, AI code assistant (copilot), AI Chat, using Wikimedia Foundation tooling, data import and export tools, generating PIDs and making deposits in a scholarly repository. # '''Data presentation and data use:''' Wikidata Query Service results, MediaWiki infoboxes, AI Chat SPARQL query processing. # '''Open Science practice:''' Open-source software, Open Notebook Science, Open Licencing, PIDs, FAIR Data Principles, and ethical and good practice AI use. ==== Sessions ==== The sessions would be about cataloguing Sprengel Museum exhibitions using LOD and how to make visualisations and presentations. '''Learning to use LOD is the goal of the learning.''' The method will be to build out from a kernel of an ‘exhibition’ and add ‘item in an exhibition’. From the start the students will be the ones who make the LOD. This will start with minimal entries my by the students, then layering these up with – Identifiers, LOD Media sources, schemas, etc. And finally moving onto how to present the data in a way that satisfies the ‘use case’: '''to gain exhibitions increased visitors numbers and create greater depth of engagement'''. Here presentation technologies are used: MediaWiki infoboxes, Wikidata Query Service results, AI Chat SPARQL queries and other features, etc. ===== Session 1: Exhibition timeline creation - build out, add exhibitions ===== # Record minimal information for an exhibition in Wikidata as Linked Open Data: Title, museum, date, etc. e.g., https://www.wikidata.org/wiki/Q138547468 – See: Table 1: ''Minimal data entries for an exhibition'' # View the exhibition record in Wikidata Query Service results link (timeline and graph https://w.wiki/J8NJ | https://w.wiki/J8aS ) # Review exhibition entries. # Cover topics raised by making a LOD entry: Wikidata basics, Wikidata good practice, consulting schemas, importance of review and using GitHub Issues, comparing available data – before and after. ===== Session 2: Exhibition cataloguing - build up, add items, artists, catalogues ===== ===== Session 3: Museum visit - Sprengel Museum ===== ===== Session 4: Schemas and Prototyping (the end of class project) ===== ===== Session 5: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 6: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 7: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 8: Prototype Creation: Data entry, visualisation, and presentation ===== --- ==== Session 1: Exhibition timeline creation - build out, add exhibitions ==== The exercise: Create a Linked Open Data record for an exhibition using Wikidata (minimal entry). A. '''Creating the exhibition entry in Wikidata.''' # Login to Wikidata: https://www.wikidata.org/ # Have a source at hand to make a data entry, e.g., #* https://www.sprengel-museum.de/ausstellungen/archiv #* https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised #* https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&currentPosition=1 # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] ''Table'' ''1: Minimal data entries for an exhibition'' {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} '''''End of Session 1.''''' ==== Homework exercises ==== # Complete your allocated exhibition. Make sure all fields are complete from Table 1. If something cannot be added, either: A. Make a note in the exhibition allocation spreadsheet, or B. Send and email to [mailto:Simon.worththington@tib.eu simon.worththington@tib.eu] and I will help resolve your issue. '''Note: If you did not create an exhibition entry during the class make sure one is complete before the next class.''' # Create a GitHub account and add your GitHub handle next to your name, column ‘GitHub handle’, in the exhibition allocation spreadsheet. # Review your classmates exhibition entries. You have all been allocated a entry to review, see the Exhibition Allocation spreadsheet. Your name will be in column G. This first review has three questions – tick the boxes to show if each item has been complete and either add comments or correct the Wikidata exhibition entry. '''Note: If your allocated Exhibition entry hasn’t been made by you classmate then please contact them and ask them to complete the entry.''' Questions are: ## Are all the required fields present? ## Are all the fields correct? ## Is there an Deutsch and English entry? --- ==== Session 2: Exhibition cataloguing - build up, add items, artists, catalogues ==== The session has five exercies: # Exhibition update # Artist # Exhibition catalogue # AI LLM SPARQL experiments # <s>Artwork</s> The exercises include the following concepts: ==== Exercises ==== ==== 1. Exhibition updates ==== * Homework review: Complete all fields for an exhibition. Review your assigned review exhibition answering the three questions: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch</blockquote> * For the label. Convert words in all caps to sentence case. Use: https://convertcase.net/title-case-converter/ | Change from, e.g., ADRIAN SAUER: TRUTH TABLESPECTRUM INTERNATIONALER PREIS FÜR FOTOGRAFIE DER STIFTUNG NIEDERSACHSEN to Adrian Sauer: Truth Tablespectrum Internationaler Preis Für Fotografie Der Stiftung Niedersachsen. * Add the English language versions. Use DeepL to translate: https://www.deepl.com/en/translator ** Title: Add English title * Add the following. Change P710 Teilnehmer (Participant) to P921 zentrales Thema '''artists name.''' ** Qualifier on central theme to indicate the person is contributing artwork. * Use: Qualifier P170 creator and add artist Q483501 (type artists and it will automcomplete) * Reference: Gemeinsame Normdatei (GND) ID for a person, e.g., Gabriela Jolowicz https://d-nb.info/gnd/134184963 | Search your persons name and copy in the last part of number 134184963 * Talk page: Add in the review questions for your Wikidata entry: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch?</blockquote>Notice the useful links that tell you more about connected Linked Open Data! Note: SPARQL query showing data model. Properties and and values. Results: https://w.wiki/JMLX Made with Gemini AI: https://gemini.google.com/share/c43f34a67f67 ==== Concepts ==== * Wikidata parts – see about and diagram: ** https://www.wikidata.org/wiki/Wikidata:Introduction/de ** https://www.wikidata.org/wiki/Wikidata:Introduction#/media/File:Datamodel_in_Wikidata.svg * Applying a review process using Talk pages * Adding References * Using a type of LOD source – '''An authority record''' Gemeinsame Normdatei (GND) ID https://portal.dnb.de/opac.htm * SPARQL query --- ==== 2. Artists ==== The objective here is to ensure all artists have been included in exhibition listing and to then review the existing artists entry. Later a SPARQL query will be made to compare statements about all the artists in our dataset. * Before reviewing artists items make sure all artists have been listed in the exhibition item, with qualifier of being an artist and a reference to their GND record. ===== Important statements ===== {| class="wikitable" |Concept |CIDOC CRM (Full) |Linked Art (Selection) |Wikidata Equivalent |Note |- |Entity |E21 Person |Person |Q5 (human) |The base instance. |- |Label/Name |P1 is identified by → E33_E41 |identified_by (Name) | |Linked Art flattens this into a simple list of names. |- | | | |P735 Given name | |- | | | |P734 Family name | |- |Profession |P2 has type → E55 Type |classified_as |P106 (occupation) |Map to AAT 300025103 (artist). |- |Birth |P98i was born → E67 Birth |born (Birth) |P569 (date of birth) |CRM treats birth as an event; Wikidata as a property. |- |Death |P100i died in → E69 Death |died (Death) |P570 (date of death) |If the artist is still living, this is omitted. |- |Nationality |P107i member of → E74 Group |classified_as (Type) |P27 (citizenship) |Linked Art often models nationality as a Type. |- |Reference |P1 identifies ← E42 Identifier |identified_by (Identifier) |QID (The URI itself) |Used to link to external authorities (ULAN, VIAF). |- |Commons category |? |? |P373 search name |<nowiki>https://commons.wikimedia.org/</nowiki> |} From Google Gemini: https://gemini.google.com/share/578cc1b886d0 --- ===== Schemas and communities need consulting. ===== From Wikimedia: * WikiProject Visual Arts: https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Visual_arts * Wikiproject Exhibitions: https://www.wikidata.org/wiki/Wikidata:WikiProject_Exhibitions Semi-formal Generic Wikibase Model for Cultural Data: https://kgi4nfdi.github.io/Guidelines/guide/wikibase/data_modelling_import/ Formal: CIDOC Conceptual Reference Model (CRM) - https://cidoc-crm.org/ Linked Art (based on CIDOC) https://linked.art/model/actor/ ==== Concepts ==== * Data modeling * Schemas * Use case * Bottom up design * Identifiers --- ==== 3. Exhibition Catalogue ==== Search in both of these two places to find information about the catalogue for your assigned exhibition. * Sprengel Museum publication catalogue - https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised * DND (example) you can search for the exhibition name or Sprengel Museum '''-''' https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true ''Note: Make a note of any links you find in the exhibition listings spreadsheet.'' ===== Make a Wikidata entry for the catalogue ===== Note: first search for publication before making Wikidata entry. Use title, use ISBN, use GND. An example publication from DNB and Sprengel Shop. * https://portal.dnb.de/opac/showFullRecord?currentResultId=Gabriela+and+Jolowicz%26any&currentPosition=0 * https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised ===== Enter these statements ===== Note: Remember Label and Description {| class="wikitable" |Property |Label |Description/Example |- |P31 |instance of |catalogue (Q2352616) |- |P1476 |title |The official title of the catalogue (e.g., Vermeer and the Masters of Genre Painting) |- |P50 |author |The main curator or art historian (item link) |- |P123 |publisher |The museum or publishing house (e.g., Louvre Museum) |- |P577 |publication date |Year of release (e.g., 2024) |- |P212 |ISBN-13 |The 13-digit standard book identifier |- | |GND |ID |- |P973 |described at URL |A link to the catalogue's page on the museum’s website |} Google Gemini https://gemini.google.com/share/9a21f5522192 Example input: https://www.wikidata.org/wiki/Q138646145 ===== Link the record back to the exhibition ===== P972 Title ==== Concepts ==== * Data modeling * Identifier * Data as CC Zero / Copyright of data --- ==== 4. AI LLM SPARQL experiments ==== The Wikidata has a SPARQL interface where the LOD in Wikidata can be searched (queried) and outputted in a number of ways, formats, and a visualisations. As well as being saved on the web. We will us AI LLM chat to generate SPARQL queries. Later we will learn the fundamentals of writing a SPARQL query. But for the moment we want to see how they have be generated, the options, and creative applications. Using chat services or code assistants can be a valuable way to learn about new technologies. {| class="wikitable" |Service |Best For |Standout Feature |Key Model(s) |- |'''ChatGPT''' |General Use & Tasks |Deep Research & Agent Mode |GPT-5.4, GPT-5 |- |'''Claude''' |Coding & Writing |Artifacts (interactive workspace) |Claude 4.5, 4.6 |- |'''Google Gemini''' |Google Ecosystem |Nano Banana (native image/video) |Gemini 3.1 Pro |- |Perplexity |Real-time Research |Native Citations & Search Labs |Sonar, GPT-5, Claude |- |MS Copilot |Office Productivity |Copilot Vision & 365 Integration |GPT-5.2, Prometheus |- |DeepSeek |Logical Reasoning |High-tier performance at low cost |DeepSeek-V3, R1 |- |Grok |Real-time Social Info |Unfiltered X (Twitter) integration |Grok 4.1 |- |'''Meta AI''' |Social Media |Seamless integration in WhatsApp/IG |Llama 4 (Scout) |- |Poe |Model Testing |Access multiple LLMs in one app |Multi-model aggregator |- |Mistral (Le Chat) |Privacy & Developers |European-hosted, GDPR-focused |Mistral Large 3 |} Some of these can also be used via KISSKI „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat  === The exercise === The group will be split into a number of Zoom breakout groups and then the group spends 20 minutes experimenting generating SPARQL queries and other creative applications. Paste in results here: https://tib.cloud/apps/files/files/8251374?dir=/NFDI4Culture/HsH/BIM26/bim26-shared&editing=false&openfile=true Each room is assigned a Chat engine. Maximum there will be four groups. ·      Group #1: '''ChatGPT''' ·      Group #2: '''Claude''' ·      Group #3: '''Google Gemini''' ·       Group #4: '''Meta AI''' === Example exercise === Chat bots can read  a SPARQL query or a Wikidata address. e.g., Item https://www.wikidata.org/wiki/Q138547468 query graph https://w.wiki/JPNc query timeline https://w.wiki/JPPN Item Sprengel Museum https://www.wikidata.org/wiki/Q510144 Then the chatbot can be instructed to do things based on the information provided. You should ask the chat bot to generate Wikidata SPARQL queries and then paste the queries into the SPARQL querie interface. https://query.wikidata.org/ Use these examples and invent your own: # Create dashboard (count of things) # Create inventory (table) # Create graph data model Some output SPARQL queries ·       Map of artists place of birth - https://w.wiki/JPT3 ·       List of exhibitions - https://w.wiki/JPR3 ·       As plot of exhibitions - https://w.wiki/J8aS ==== Homework: Session 2 ==== Create a bottom up data model of an artwork in an exhibition. Include only the minimum information needed. The result should be a table like the ones presented for exhibition, artist, and catalogue. The table should include properties and attributes. You should consult the schemas mentioned above. You can use AI but attribute the AI and link to your question. If you use AI review the results and make notes about what you changed. Note: Think about how parts are related and what you need to add and what already exists in Wikidata. Submit your results as a spreadsheet or table. ===== Session 3: Museum visit - Sprengel Museum ===== 19 March 2026 ===== Session #4: Schemas and Prototyping (the end of class project) ===== ===== Recap and outline ===== Done * Creating exhibition entries in Wikidata * Filling our data models for Artist and Catalogue * Exploring the museum and its activities to help steer the prototype To do * Decide on the ideas for the prototype * Data model for items in an exhibition (Artwork and Exhibition) * Complete a data model for the end of the project that can be used by museums and complies to the sector standards – CIDOC and Wikidata. ===== What have we learned about the ‘Museum’s Story’ ===== TBC ===== Schemas ===== An opportunity to become familiar with how Linked Open Data is structured using common agreements on working practices. Over the period of the course a data model will be developed and finalised to describe ‘items in an exhibition’. The data model will be published for community consultation and testing. ===== Schemas and key concepts ===== Table: https://tib.cloud/s/ZKNAAo3B8ATXsAP * Schema * Terminology Service * Controlled Vocabulary * Taxonomy * Ontology * Knowledge Graph Table X: link: https://tib.cloud/s/ZKNAAo3B8ATXsAP Terminology used in Linked Open Data (LOD) {| class="wikitable" |- ! **Concept** ! **Wikidata link (Concept)** ! **Primary Focus** ! **Analogy** ! **Example resource** ! **URL** ! **Example use** ! **URL** |- | Schema | Q1397073 | Data Structure | The Template. Conceptual schema / data model | Schema.org | https://schema.org/ | VisualArtwork | https://schema.org/VisualArtwork |- | | | | | | | Smithsonian American Art Museum (SAAM) "Among the Sierra Nevada, California" | https://www.wikidata.org/wiki/Q20475372 |- | Terminology Service | Q22692845 | Distribution | A Library of Vocabularies, Schemas, Ontologies, etc | TIB Terminology Service | https://terminology.tib.eu/ts/ | NFDI4CULTURE | https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE |- | Controlled Vocabulary | Q1469824 | Consistency | The Dictionary | Integrated Authority File / die Gemeinsame Normdatei (GND) | https://portal.dnb.de/opac/showShortList | Persons: Dürer, Albrecht | https://d-nb.info/gnd/117751669 |- | Taxonomy | Q8269924 | Hierarchy | Sorting things by type (general classification) | Getty Art & Architecture Thesaurus (AAT) | https://www.getty.edu/research/tools/vocabularies/aat/ | German Surrealist Max Ernst (painting techniques used) | https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20and%20Grattage,in%20his%20drawings%20in%201925. |- | | | | | Iconclass | https://iconclass.org/ | Max Ernst’s "The Virgin Spanking the Christ Child" (Parady) | https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926 |- | Ontology | Q324254 | Semantics: Meaning & logic (information science) | The Rulebook or Writing Style Guide | CIDOC (Comité International pour la DOCumentation / International Committee for Documentation) | https://cidoc-crm.org/ | Sloane Lab Knowledge Base - unifying 3 collections | https://knowledgebase.sloanelab.org/resource/Start |- | Knowledge Graph | Q33002955 | Network of things and relations | A Navigational Map | Census of Antique Works of Art and Architecture Known in the Renaissance | https://www.census.de/ | Artemis search | https://database.census.de/#/detail/10013099 |- | | | | | Research Space | https://researchspace.org/ | Hokusai: The Great Picture Book of Everything | https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start |} ===== Schemas exercise ===== Spreadsheet to work on: https://tib.cloud/s/PicTdwCEqCQ6pBp (password: bim2026) We will be looking at: Exhibition, Artist, and Catalogue. '''''Enter the URLs found. Add new rows, columns, comments if needed. Keep manual searches as well as AI searches for comparison.''''' ===== Exercise #1: Enter links into the spreadsheet of matching items from the following: ===== * Wikidata:WikiProject Exhibitions/Properties * Generic Wikibase Model for Cultural Data - Wikibase4Research NFDI4Culture * CIDOC CRM (Full) * Terminology Service (NFDII4Culture) * Wikidata ===== Exercise #2: Use AI LLM to find matching items ===== * https://gemini.google.com/ ==== Prototyping ==== Either in this session or in the next session the group will be divided into teams. ===== Schema ===== # Data model development: ‘items in an exhibition’ ===== Quarto publication parts ===== # A catalogue of a Sprengel Museum exhibition # A catalogue of all exhibitions and exhibition catalogues # Catalogue of exhibition entries --- ==== Learning to use Quarto and inserting an exhibition entry ==== Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype '''Model: Auto''' '''How the repo was setup. Agent promts:'''<blockquote>''I want to run a Quarto website project, please setup the basics. The project will be published on GitHub Pages. Set the output directory to docs.'' </blockquote>Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minutes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- ===== Homework - session #4 ===== * Get all books from HsH library that are Sprengel Museum exhibition catalogues. Bring to the next class * Make an exhibition entry if not done * Work with VSCode and the Agent and experiment * Add entries from existing ontologies: https://tib.cloud/s/PicTdwCEqCQ6pBp?dir=/&editing=false&openfile=true ==== Sitzung 5: Prototyping: Running Quarto Prototype, Federation, Prototype Teams ==== * Running Quarto Prototype - https://github.com/mrchristian/prototype * DNB data download https://github.com/NFDI4Culture/linked-open-exhibition * Data Federation - WB4R ===== Links ===== https://wikibase.wbworkshop.tibwiki.io/wiki/Main_Page Glossar - https://nfdi4culture.github.io/linked-open-exhibition/Documentation/glossary ===== DNB Search ===== https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books Sprengel Museum, 602 Artikel '''About the DNB''' https://www.dnb.de/librarylab https://deutsche-nationalbibliothek.github.io/jupyterlite/lab/ '''Also''' https://wiki.dnb.de/spaces/LINKEDDATASERVICE/pages/449878933/DNB+SPARQL+Service+BETA ===== Prototype Teams ===== * DNB Data * Katalog scan * Exhibition entries * Datenmodell (alle) ===== --- ===== == Session 6: Class Project – Prototyping: ''Linked Open Exhibitions'' == Prototype URL (currently 2026-04-29 a shell framework): https://nfdi4culture.github.io/linked-open-exhibition/ ==== Program: ==== 11:30 – 11:50 (20 min) '''Outline: Class Project – Prototyping: Linked Open Exhibitions.''' What is it and what needs to be delivered. Allocation to sub-project and tasks outline. '''Activity #1: Bottom-up data modeling: Data mapping''' 11:50 – 12:20 (30 min) Data finding and exploration (break out rooms) 12:20 – 12:40 (20 min) Review data findings (class discussion) 12:40 – 12:55 (15 min) Pause Break '''Activity #2: Top-down data modeling: Schema mapping''' 12:55 – 13:35 (40 min) Map data against schemas (break out rooms) 13:35 – 13:55 (20 min) Review findings (class discussion) '''13:55 – 14:15 (20 min) Work time: Open time-slot to review running ‘Tech Stack’ or address any other questions''' --- ==== Important links ==== * '''Main Prototype repository:''' https://nfdi4culture.github.io/linked-open-exhibition/ * Quarto setup: ‘''BIM Prototype 02 Quarto Website’:'' https://mrchristian.github.io/prototype/ * Instructions for ‘Tech Stack’: ''Einführung in Quarto und Einfügen eines Ausstellungsbeitrags'' [[BIM-126-02-Data-Science-Linked-Open-Exhibition#Einführung in Quarto und Einfügen eines Ausstellungsbeitrags|https://en.wikiversity.org/wiki/BIM-126-02-Data-Science-Linked-Open-Exhibition#Einf%C3%BChrung_in_Quarto_und_Einf%C3%BCgen_eines_Ausstellungsbeitrags]] * Earlier Prototype (2025): https://nfdi4culture.github.io/open-museum/ ==== ''Outline: Class Project – Prototyping: Linked Open Exhibitions'' ==== Prototype: https://nfdi4culture.github.io/linked-open-exhibition/ Repo: https://nfdi4culture.github.io/linked-open-exhibition/ Why? * Rapid Prototyping in this context is used to learn about ‘Data Modeling using Linked Open Data’. '''''NB: The data modeling skills and experience learned here is a core competence that gives a foundation to be able to create data models in a wide set of professional contexts.''''' ** How to do data modeling ** To use method: Bottom-up; KISS (Keep it Short and Simple); Top-down ** Evaluation and validation ** Operationalize a data model ** User testing ** Good practice, including Open Scholarship (Open Science) practice. e.g. FAIR Data Principles ** Experiment with AI LLMs and agentic coding in the workflow * Rapid Prototyping is a Design Research methodology – meaning to create or discover knowledge by doing. What? Create a website prototype a the whole class: https://nfdi4culture.github.io/linked-open-exhibition/ The website is made of three data driven sub-projects: # Manual Wikidata entries for Sprengel Museum website – class entries already made # Bulk exhibition entries derived from 600+ DNB records for ‘Sprengel Museum’ - imported # HsH Library information records for a search on the Sprengel Museum and one scan of a Sprengel Museum catalogue for Text and Data Mining (TDM) – to do How? Simon Worthington will act as Publication Manager. This involves running or guiding complex software parts. Copilot agentic coding will be used (experimented with) for some parts. Class is divided into three teams of the sub-projects: # Sprengel Museum exhibitions website; # DNB records ‘Sprengel Museum’ # Text and Data Mining: Library catalogue Sprengel Museum Each team carries out the same tasks for their parts to complete a round of data modeling: # Collect data – bottom-up method # Validate the data – top-down method # Presentation of data – Quarto website ‘[https://nfdi4culture.github.io/linked-open-exhibition/ Linked Open Exhibitions]’ Goal: Definition-of-done (DoD) ''NB: Developer speak'' * A documented data model (Table) with diagram (Mermaid, GraphVis, or Draw.io) * Mapping of data model to schemas (Table) * Idea for presentation of data for each sub-section in the prototype and implementation with Publication Manager assistance, e.g., for DNB a chronological list of exhibitions with images. * Documentation of AI LLM use as an assistant, attribution, and comments on good practice * Data provenance and good practice checklist completion * The final result ‘Class Project – Prototyping: Linked Open Exhibitions’ will be made as an institutional deposit with [https://zenodo.org/ Zenodo]. --- ==== Activity #1: Collecting data and bottom up data modeling ==== Confirm, create, or expand existing data models by looking at the source. Each project has a source: * Team #1: Sprengel Museum website, exhibition listings: ** https://www.sprengel-museum.de/ and ** spreadsheet of entered exhibitions CSV GitHub | Spreadsheet TIB Cloud (passworded) ** In prototype: https://nfdi4culture.github.io/linked-open-exhibition/exhibitions.html * Team #2: DNB records of search for ‘Sprengel Museum’: ** https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books | https://wikibase.wbworkshop.tibwiki.io/ ** CSV https://github.com/NFDI4Culture/linked-open-exhibition/blob/main/catalogues/sprengel_exhibitions.csv ** Images of book covers https://github.com/NFDI4Culture/linked-open-exhibition/tree/main/catalogues/images * Team #3: HsH Library information on catalogues for ‘Sprengel Museum’: ** https://katalog.bib.hs-hannover.de/vufind/Search/Results?lookfor=Sprengel%2BMuseum ** Team 3 have to start from scratch as as yet we don’t have a library record or item in an exhibition data model. Tip: look back at the other models to start building up your data models. '''>>> Add to data model here:''' https://tib.cloud/s/PicTdwCEqCQ6pBp (passworded) ==== OBJECTIVE ==== To ensure that the data model can represent the source. Are there enough entries to describe the things that make up the source. The process is iterative, meaning it keeps on being repeated with improvements and changes being made. ==== TASKS ==== * Edit the purple grey area, thje green are will be edited in the next activity * Review and correct existing information * Add new concepts if the source needs it * Orange areas need filling in. The cells might need editing or added to. * Data types can be found on Property Pages only Items (QIDs) don’t have data types, in <nowiki>https://www.wikidata.org/wiki/Property:P1476</nowiki> under the Label - ''Data type'' * URI is equivelent to URL # Tips * Look at other examples on Wikidata: Artists, exhibitions, catalogues, bibliographic, or items in an exhibition. * Use an AI to lookup schema explanations or options. Register with KISSKI to get better AI privacy use. ## Activity #2: Top down data modeling validation * Team #1: Sprengel Museum website * Team #2: DNB records of search for ‘Sprengel Museum’ * Team #3: HsH Library information on catalogues for ‘Sprengel Museum’ # OBJECTIVE Map all concepts ==== TASKS ==== Look up the concept in the different resources and add mapping links, ## Work time: Open time-slot to review running ‘Tech Stack’ or address any other questions --- ## Homework * Complete the Bottom-up and Top-down modelling * Team #3: Visit the library and make a digital scan on a copier machine, store as PDF. The scan will be used for text and data mining and the file deleted and destroyed after. We will only be extracting metadata from the scan. * Come along to the next class with ideas and suggestions for what you would like to have displayed from your data and data models in the prototype. [[Category:Wikidata]] 3ezsvba2km6o9qmgkfht4pvds28oidu 2807036 2807035 2026-04-29T17:14:48Z Mrchristian 281704 /* TASKS */ 2807036 wikitext text/x-wiki DE (EN Below) {{TOCleft}} ==== Linked Open Exhibition ==== ''Materialien und Aufgaben für das Modul „BIM-126-02, SoSe 2026, Worthington/Blümel” für Studierende der Hochschule Hannover. Die Materialien werden gemeinsam mit mehreren Kollegen aus dem [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab] der TIB Hannover erstellt.'' Projekt-GitHub-Repo: https://github.com/NFDI4Culture/linked-open-exhibition ==== Zusammenfassung ==== Der achtteilige Kurs bietet eine Einführung in Linked Open Data (LOD) im Kontext von: # Open Galleries Libraries Archives and Museums (GLAM) und # der Nutzung von Plattformen der Wikimedia Foundation. Die folgenden Plattformen der Wikimedia Foundation werden verwendet: Wikidata, Wikibase, MediaWiki und Wikimedia Commons. AI LLM wird in den folgenden Workflows verwendet: Code Assistant ''Copilot'' und eine Vielzahl von AI LLM-Chat-Diensten für die Dateierstellung und Konfigurationen zur Erstellung von SPARQL-Abfragen, Jinja 2.0-Vorlagen usw. Das „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) kann für unbegrenztes ChatGPT5 genutzt werden: https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat Die verwendeten Methoden sind: Open-Source-Software, Open Science und Rapid Prototyping. ==== Linked Open Exhibition ==== Die Frage, die in diesem Kurs untersucht wird, lautet: Wie kann LOD genutzt werden, um Museumsausstellungen als Linked Open Exhibitions zu verbessern – als Aufzeichnung der Ausstellung, als Katalog der Ausstellungsstücke und für andere wichtige Daten? Als Beispiele '''dienen die Steigerung der Besucherzahlen von Ausstellungen und die Schaffung einer größeren Tiefe des Engagements'''. Der Schwerpunkt liegt auf der Frage, wie LOD-Aufzeichnungen von '''Exponaten in einer Ausstellung''' erstellt werden können. ==== Lernpunkte – in der Reihenfolge ihrer Priorität ==== # '''Wikidata/Wikibase LOD-Konzepte:''' Objekte, Eigenschaften, Werte, Qualifikatoren, Wikibase-Schemas, Klassen, Lexeme, Wissensbasis und Wissensgraphen. # '''Linked Open Data (LOD):''' Semantic Web, 5-Sterne-Bewertung, RDF/Triples, Ontologien, Taxonomien und kontrollierte Vokabulare. # '''Verwendung von LOD-Quellen:''' Identifikatoren, PIDs, Informationsquellen, Medienquellen sowie Import- und Export-Tools. # '''Datenmodellierung:''' Methodiken, Schemaverwendung, Visualisierung und Testen. # '''Daten-Workflow-Tools:''' Git, IDE, KI-Code-Assistent (Copilot), KI-Chat, Verwendung von Wikimedia Foundation-Tools, Datenimport- und -export-Tools, Generierung von PIDs und Hinterlegung in einem wissenschaftlichen Repositorium. # '''Datenpräsentation und Datennutzung:''' Ergebnisse des Wikidata Query Service, MediaWiki-Infoboxen, Verarbeitung von SPARQL-Abfragen durch KI-Chat. # '''Open-Science-Praxis:''' Open-Source-Software, Open Notebook Science, Open Licensing, PIDs, FAIR-Datenprinzipien sowie ethische und bewährte Verfahren bei der Nutzung von KI. ==== Sitzungen ==== Die Sitzungen befassen sich mit der Katalogisierung von Ausstellungen des Sprengel Museums unter Verwendung von LOD und der Erstellung von Visualisierungen und Präsentationen. '''Das Ziel des Lernens ist es, den Umgang''' mit '''LOD''' zu '''erlernen.''' Die Methode besteht darin, ausgehend von einem Kern einer „Ausstellung” „Exponate in einer Ausstellung” hinzuzufügen. Von Anfang an sind es die Studierenden, die die LOD erstellen. Dies beginnt mit minimalen Einträgen der Studierenden, die dann mit Identifikatoren, LOD-Medienquellen, Schemata usw. ergänzt werden. Schließlich wird gezeigt, wie die Daten so präsentiert werden können, dass sie dem „Anwendungsfall” entsprechen: '''die Besucherzahlen der Ausstellungen zu steigern und ein tieferes Engagement zu erreichen'''. Hier kommen Präsentationstechnologien zum Einsatz: MediaWiki-Infoboxen, Ergebnisse des Wikidata Query Service, KI-Chat-SPARQL-Abfragen und andere Funktionen usw. ==== Sitzung 1: Erstellung einer Ausstellung-Zeitleiste – Aufbau, Hinzufügen von Ausstellungen ==== # Erfassen Sie minimale Informationen zu einer Ausstellung in Wikidata als Linked Open Data: Titel, Museum, Datum usw. Beispiel: https://www.wikidata.org/wiki/Q138547468 – Siehe: Tabelle1 : ''Minimale Dateneinträge für eine Ausstellung'' # Zeigen Sie den Ausstellungseintrag in Wikidata an Ergebnisse des Abfragedienstes anzeigen Link (Zeitleiste und Grafik https://w.wiki/J8NJ | https://w.wiki/J8aS ) # Überprüfen Sie die Ausstellungseinträge. # Behandeln Sie Themen, die durch die Erstellung eines LOD-Eintrags aufgeworfen werden: Wikidata-Grundlagen, bewährte Verfahren für Wikidata, Konsultation von Schemata, Bedeutung der Überprüfung und Verwendung von GitHub Issues, Vergleich der verfügbaren Daten – vorher und nachher. ==== Sitzung 2: Ausstellungskatalogisierung – Aufbau, Hinzufügen von Objekten, Künstlern, Katalogen ==== ==== Sitzung 3: Museumsbesuch – Sprengel Museum (noch zu bestätigen) ==== ==== Sitzung 4: Ausstellungskatalogisierung – Massenhinzufügungen: Hinzufügen von Objekten, Künstlern, Katalogen ==== ==== Sitzung 5: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 6: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 7: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 8: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== --- ==== Sitzung 1: Erstellung eines Ausstellungskalenders – Aufbau, Hinzufügen von Ausstellungen ==== Die Übung: Erstellen Sie einen Linked-Open-Data-Datensatz für eine Ausstellung mit Wikidata (Mindestangaben). A. '''Erstellen des Ausstellungseintrags in Wikidata.''' # Anmeldung bei Wikidata: https://www.wikidata.org/ # Halten Sie eine Quelle bereit, um Daten einzugeben, z. B. #* https://www.sprengel-museum.de/ausstellungen/archiv #* https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised #* https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&currentPosition=1 # Überprüfen Sie, ob es bereits einen Eintrag für die Ausstellung auf Wikidata gibt. Verwenden Sie dazu die Suchfunktion. # Erstellen Sie einen Eintrag oder bearbeiten Sie einen bestehenden Eintrag. #* Hinweis: Überprüfen Sie, welche Sprache Sie verwenden. Wir werden Einträge in Deutsch und Englisch hinzufügen (beginnend mit Deutsch). # Erstellen Sie die folgenden Dateneinträge in Wikidata, siehe: Tabelle 1: ''Minimale Dateneinträge für eine Ausstellung.'' # Überprüfen Sie die Wikidata-Einträge zur Ausstellung. Die Überprüfung erfolgt anhand von drei Fragen. Fügen Sie bei Bedarf Kommentare hinzu, Korrekturen können vorgenommen werden. Ergebnisse und Anmerkungen können auf der Diskussionsseite des Eintrags hinzugefügt werden, z. B. #* Alle Einträge vorhanden [ ] #* Alle Einträge sind korrekt [ ] #* Einträge sind in Deutsch und Englisch – im Rahmen des Zumutbaren [ ] ''Tabelle2 : Mindestdaten für einen Ausstellungseintrag'' {| class="wikitable" | colspan="7" |'''Felder, die zur Erstellung eines Ausstellungseintrags verwendet werden. Siehe Beispiel: https://www.wikidata.org/wiki/Q138547468''' |- |A |Beschriftung | colspan="5" |Hinweis: Kurz halten. Titel der Ausstellung verwenden |- |B |Beschreibung | colspan="5" |Hinweis: Zur Unterscheidung von anderen Einträgen verwenden. Folgen Sie diesem Beispiel: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Eigentum (P) und Objekt (Q)''' |'''URI''' |'''DE''' | |'''Hinzufügen''' |'''Anmerkung''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |Instanz von |Q464980 |Element hinzufügen |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Ausstellung | |(oben verwendet) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Titel |Titel |Klartext |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Standort |Sprengel Museum Hannover Q510144 |Artikel hinzufügen |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Startzeit |Datum |JJJJ-MM-TT |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |Endzeit |Datum |JJJJ-MM-TT |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Kurator |Person |Element hinzufügen (falls nicht vorhanden, muss erstellt werden/kann derzeit weggelassen werden) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Teilnehmer |Person (der Künstler) |Element hinzufügen (falls nicht vorhanden, muss erstellt werden/kann derzeit weggelassen werden) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |Offizielle Website |Offizielle Website |URL |URL |} Ende von Sitzung 1. ==== Hausaufgabenübungen ==== 1. Vervollständigen Sie Ihre zugewiesene Ausstellung. Stellen Sie sicher, dass alle Felder aus Tabelle 1 ausgefüllt sind. Wenn etwas nicht hinzugefügt werden kann, haben Sie zwei Möglichkeiten: A. Machen Sie eine Notiz in der Tabelle zur Ausstellungszuweisung oder B. Senden Sie eine E-Mail an [mailto:Simon.worththington@tib.eu simon.worththington@tib.eu] , damit ich Ihnen bei der Lösung Ihres Problems helfen kann. '''Hinweis: Wenn Sie während des Unterrichts keinen Ausstellungseintrag erstellt haben, stellen Sie sicher, dass dieser vor der nächsten Unterrichtsstunde fertiggestellt ist.''' 2. Erstellen Sie ein GitHub-Konto und fügen Sie Ihren GitHub-Namen neben Ihrem Namen in der Spalte „GitHub-Name” in der Tabelle zur Zuweisung der Ausstellungen hinzu. 3. Überprüfen Sie die Ausstellungseinträge Ihrer Klassenkameraden. Ihnen wurde allen ein Eintrag zur Überprüfung zugewiesen, siehe Tabelle zur Zuweisung der Ausstellungen. Ihr Name steht in Spalte G. Diese erste Überprüfung umfasst drei Fragen – kreuzen Sie die Kästchen an, um anzuzeigen, ob jeder Punkt ausgefüllt wurde, und fügen Sie entweder Kommentare hinzu oder korrigieren Sie den Wikidata-Ausstellungseintrag. '''Hinweis: Wenn der Ihnen zugewiesene Ausstellungseintrag nicht von Ihrem Klassenkameraden erstellt wurde, kontaktieren Sie ihn bitte und bitten Sie ihn, den Eintrag zu vervollständigen.''' Die Fragen lauten: 1. Sind alle erforderlichen Felder vorhanden? 2. Sind alle Felder korrekt ausgefüllt? 3. Gibt es einen deutschen und einen englischen Eintrag? --- === Sitzung 2: Ausstellungskatalogisierung – Aufbau, Hinzufügen von Objekten, Künstlern, Katalogen === ==== Die Sitzung umfasst fünf Übungen: ==== # Ausstellungsaktualisierung # Künstler # Ausstellungskatalog # AI LLM SPARQL-Experimente # <s>Kunstwerk</s> ==== Die Übungen umfassen die folgenden Konzepte: ==== ==== Übungen ==== ==== 1. Aktualisierung der Ausstellung ==== * Hausaufgabenüberprüfung: Füllen Sie alle Felder für eine Ausstellung aus. Überprüfen Sie die Ihnen zugewiesene Ausstellung, indem Sie die folgenden drei Fragen beantworten: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch</blockquote> * Für das Label. Wandeln Sie Wörter in Großbuchstaben in Satzschrift um. Verwenden Sie: https://convertcase.net/title-case-converter/ | Ändern Sie z. B. ADRIAN SAUER: TRUTH TABLESPECTRUM INTERNATIONALER PREIS FÜR FOTOGRAFIE DER STIFTUNG NIEDERSACHSEN in Adrian Sauer: Truth Tablespectrum Internationaler Preis Für Fotografie Der Stiftung Niedersachsen. * Fügen Sie die englischen Versionen hinzu. Verwenden Sie DeepL zum Übersetzen: https://www.deepl.com/en/translator ** Titel: Fügen Sie den englischen Titel hinzu * Fügen Sie Folgendes hinzu. Ändern Sie P710 Teilnehmer (Participant) in P921 zentrales Thema artists name. ** Qualifier zum zentralen Thema, um anzugeben, dass die Person Kunstwerke beisteuert. * Verwenden Sie: Qualifier P170 creator und fügen Sie artist Q483501 hinzu (geben Sie „Künstler” ein, es wird automatisch vervollständigt) * Referenz: Gemeinsame Normdatei (GND) ID für eine Person, z. B. Gabriela Jolowicz https://d-nb.info/gnd/134184963 | Suchen Sie den Namen der Person und kopieren Sie den letzten Teil der Nummer 134184963 * Diskussionsseite: Fügen Sie die Überprüfungsfragen für Ihren Wikidata-Eintrag hinzu: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch?</blockquote>Beachten Sie die nützlichen Links, die Ihnen mehr über verbundene Linked Open Data verraten! Hinweis: SPARQL-Abfrage zur Anzeige des Datenmodells. Eigenschaften und Werte. Ergebnisse: https://w.wiki/JMLX Erstellt mit Gemini AI: https://gemini.google.com/share/c43f34a67f67 ==== Konzepte ==== * Wikidata-Teile – siehe Informationen und Diagramm: ** https://www.wikidata.org/wiki/Wikidata:Introduction/de ** https://www.wikidata.org/wiki/Wikidata:Introduction#/media/File:Datamodel_in_Wikidata.svg * Anwendung eines Überprüfungsprozesses mithilfe von Diskussionsseiten * Hinzufügen von Referenzen * Verwendung einer LOD-Quelle – Ein Normdatensatz Gemeinsame Normdatei (GND) ID <nowiki>https://portal.dnb.de/opac.htm</nowiki> * SPARQL-Abfrage --- ==== 2. Künstler ==== Das Ziel hierbei ist es, sicherzustellen, dass alle Künstler in die Ausstellungsliste aufgenommen wurden, und anschließend die bestehenden Künstlereinträge zu überprüfen. Später wird eine SPARQL-Abfrage durchgeführt, um Aussagen über alle Künstler in unserem Datensatz zu vergleichen. Bevor Sie die Künstereinträge überprüfen, stellen Sie sicher, dass alle Künstler im Ausstellungseintrag aufgeführt sind, mit dem Qualifikationsmerkmal „Künstler” und einem Verweis auf ihren GND-Datensatz. ==== Wichtige Aussagen ==== {| class="wikitable" |Concept |CIDOC CRM (Full) |Linked Art (Selection) |Wikidata Equivalent |Note |- |Entity |E21 Person |Person |Q5 (human) |The base instance. |- |Label/Name |P1 is identified by → E33_E41 |identified_by (Name) | |Linked Art flattens this into a simple list of names. |- | | | |P735 Given name | |- | | | |P734 Family name | |- |Profession |P2 has type → E55 Type |classified_as |P106 (occupation) |Map to AAT 300025103 (artist). |- |Birth |P98i was born → E67 Birth |born (Birth) |P569 (date of birth) |CRM treats birth as an event; Wikidata as a property. |- |Death |P100i died in → E69 Death |died (Death) |P570 (date of death) |If the artist is still living, this is omitted. |- |Nationality |P107i member of → E74 Group |classified_as (Type) |P27 (citizenship) |Linked Art often models nationality as a Type. |- |Reference |P1 identifies ← E42 Identifier |identified_by (Identifier) |QID (The URI itself) |Used to link to external authorities (ULAN, VIAF). |- |Commons category |? |? |P373 search name |<nowiki>https://commons.wikimedia.org/</nowiki> |} Aus Google Gemini: https://gemini.google.com/share/578cc1b886d0 --- ==== Schemas und Communities benötigen Beratung. ==== '''Aus Wikimedia:''' WikiProject Visual Arts: https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Visual_arts Wikiproject Exhibitions: https://www.wikidata.org/wiki/Wikidata:WikiProject_Exhibitions '''Halbformell''' Generisches Wikibase-Modell für Kulturdaten: https://kgi4nfdi.github.io/Guidelines/guide/wikibase/data_modelling_import/ '''Formell:''' CIDOC Conceptual Reference Model (CRM) – https://cidoc-crm.org/ Linked Art (basierend auf CIDOC) https://linked.art/model/actor/ ==== Konzepte ==== * Datenmodellierung * Schemas * Anwendungsfall * Bottom-up-Design * Identifikatoren --- ==== 3. Ausstellungskatalog ==== Suchen Sie an beiden Orten nach Informationen zum Katalog Ihrer zugewiesenen Ausstellung. Sprengel Museum Publikationskatalog – https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised DND (Beispiel) Sie können nach dem Namen der Ausstellung oder dem Sprengel Museum suchen – https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true Hinweis: Notieren Sie sich alle Links, die Sie in der Tabelle mit den Ausstellungslisten finden. ===== Erstellen Sie einen Wikidata-Eintrag für den Katalog. ===== Hinweis: Suchen Sie zunächst nach der Veröffentlichung, bevor Sie einen Wikidata-Eintrag erstellen. Verwenden Sie den Titel, die ISBN und die GND. Ein Beispiel für eine Veröffentlichung aus DNB und Sprengel Shop. * https://portal.dnb.de/opac/showFullRecord?currentResultId=Gabriela+and+Jolowicz%26any&currentPosition=0 * https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised ===== Geben Sie diese Angaben ein ===== Hinweis: Denken Sie an die Bezeichnung und Beschreibung {| class="wikitable" |Property |Label |Description/Example |- |P31 |instance of |catalogue (Q2352616) |- |P1476 |title |The official title of the catalogue (e.g., Vermeer and the Masters of Genre Painting) |- |P50 |author |The main curator or art historian (item link) |- |P123 |publisher |The museum or publishing house (e.g., Louvre Museum) |- |P577 |publication date |Year of release (e.g., 2024) |- |P212 |ISBN-13 |The 13-digit standard book identifier |- | |GND |ID |- |P973 |described at URL |A link to the catalogue's page on the museum’s website |} Google Gemini https://gemini.google.com/share/9a21f5522192 Beispiel für eine Eingabe: https://www.wikidata.org/wiki/Q138646145 ==== Verlinken Sie den Datensatz zurück zur Ausstellung. ==== P972 > Titel ==== Konzepte ==== * Datenmodellierung * Identifikator * Daten als CC Zero / Urheberrecht der Daten --- ==== 4. AI LLM SPARQL-Experimente ==== Wikidata verfügt über eine SPARQL-Schnittstelle, über die die LOD in Wikidata durchsucht (abgefragt) und auf verschiedene Arten, in verschiedenen Formaten und Visualisierungen ausgegeben werden kann. Außerdem kann sie im Web gespeichert werden. Wir werden den AI LLM-Chat verwenden, um SPARQL-Abfragen zu generieren. Später werden wir die Grundlagen des Schreibens einer SPARQL-Abfrage lernen. Aber zunächst wollen wir sehen, wie sie generiert werden, welche Optionen es gibt und wie sie kreativ eingesetzt werden können. Die Verwendung von Chat-Diensten oder Code-Assistenten kann eine wertvolle Möglichkeit sein, um neue Technologien kennenzulernen. {| class="wikitable" |Service |Best For |Standout Feature |Key Model(s) |- |'''ChatGPT''' |General Use & Tasks |Deep Research & Agent Mode |GPT-5.4, GPT-5 |- |'''Claude''' |Coding & Writing |Artifacts (interactive workspace) |Claude 4.5, 4.6 |- |'''Google Gemini''' |Google Ecosystem |Nano Banana (native image/video) |Gemini 3.1 Pro |- |Perplexity |Real-time Research |Native Citations & Search Labs |Sonar, GPT-5, Claude |- |MS Copilot |Office Productivity |Copilot Vision & 365 Integration |GPT-5.2, Prometheus |- |DeepSeek |Logical Reasoning |High-tier performance at low cost |DeepSeek-V3, R1 |- |Grok |Real-time Social Info |Unfiltered X (Twitter) integration |Grok 4.1 |- |'''Meta AI''' |Social Media |Seamless integration in WhatsApp/IG |Llama 4 (Scout) |- |Poe |Model Testing |Access multiple LLMs in one app |Multi-model aggregator |- |Mistral (Le Chat) |Privacy & Developers |European-hosted, GDPR-focused |Mistral Large 3 |} Einige davon können auch über KISSKI genutzt werden. Das „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) kann für unbegrenztes ChatGPT5 genutzt werden: https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat ==== Die Übung ==== Die Gruppe wird in mehrere Zoom-Breakout-Gruppen aufgeteilt und verbringt dann 20 Minuten damit, SPARQL-Abfragen und andere kreative Anwendungen zu generieren. Fügen Sie die Ergebnisse hier ein: https://tib.cloud/apps/files/files/8251374?dir=/NFDI4Culture/HsH/BIM26/bim26-shared&editing=false&openfile=true Jedem Raum wird eine Chat-Engine zugewiesen. Es gibt maximal vier Gruppen. · Gruppe Nr. 1: ChatGPT · Gruppe Nr. 2: Claude · Gruppe Nr. 3: Google Gemini · Gruppe Nr. 4: Meta AI ==== Beispielübung ==== Chatbots können eine SPARQL-Abfrage oder eine Wikidata-Adresse lesen. z. B. * Artikel https://www.wikidata.org/wiki/Q138547468 * Abfragegrafik https://w.wiki/JPNc * Abfragetidsachse https://w.wiki/JPPN * Artikel Sprengel Museum https://www.wikidata.org/wiki/Q510144 Anschließend kann der Chatbot angewiesen werden, auf Grundlage der bereitgestellten Informationen bestimmte Aktionen auszuführen. Sie sollten den Chatbot bitten, Wikidata-SPARQL-Abfragen zu generieren, und diese Abfragen dann in die SPARQL-Abfrageoberfläche einfügen. https://query.wikidata.org/ Verwenden Sie diese Beispiele und entwickeln Sie Ihre eigenen: # Dashboard erstellen (Anzahl der Dinge) # Inventar erstellen (Tabelle) # Graphdatenmodell erstellen Einige SPARQL-Abfragen · Karte der Geburtsorte von Künstlern – https://w.wiki/JPT3 · Liste der Ausstellungen – https://w.wiki/JPR3 · Als Darstellung der Ausstellungen – https://w.wiki/J8aS ==== Hausaufgabe: Sitzung 2 ==== Erstellen Sie ein Bottom-up-Datenmodell eines Kunstwerks in einer Ausstellung. Fügen Sie nur die minimal erforderlichen Informationen hinzu. Das Ergebnis sollte eine Tabelle sein, wie sie für Ausstellung, Künstler und Katalog dargestellt wird. Die Tabelle sollte Eigenschaften und Attribute enthalten. Sie sollten die oben genannten Schemata zu Rate ziehen. Sie können KI verwenden, aber geben Sie die KI an und verlinken Sie sie mit Ihrer Frage. Wenn Sie KI verwenden, überprüfen Sie die Ergebnisse und machen Sie sich Notizen darüber, was Sie geändert haben. Hinweis: Überlegen Sie, wie die Teile miteinander in Beziehung stehen, was Sie hinzufügen müssen und was bereits in Wikidata vorhanden ist. Reichen Sie Ihre Ergebnisse als Tabelle oder Spreadsheet ein. --- ==== Sitzung 3: Museumsbesuch – Sprengel Museum ==== 19 März 2026 ENDE ==== Sitzung Nr. 4: Schemata und Prototyping (Abschlussprojekt) ==== ===== Zusammenfassung und Überblick ===== Erledigt * Erstellen von Ausstellungs-Einträgen in Wikidata * Befüllen unserer Datenmodelle für „Künstler“ und „Katalog“ * Erkundung des Museums und seiner Aktivitäten, um den Prototyp zu steuern Zu erledigen * Entscheidung über die Ideen für den Prototyp * Datenmodell für Objekte in einer Ausstellung (Kunstwerk und Ausstellung) * Erstellen eines Datenmodells zum Projektende, das von Museen genutzt werden kann und den Branchenstandards – CIDOC und Wikidata – entspricht. ===== Was haben wir über die „Geschichte des Museums“ gelernt? ===== TBC ===== Schemas ===== Eine Gelegenheit, sich mit der Struktur von Linked Open Data anhand gemeinsamer Vereinbarungen zu Arbeitspraktiken vertraut zu machen. Im Laufe des Kurses wird ein Datenmodell entwickelt und fertiggestellt, um „Objekte in einer Ausstellung“ zu beschreiben. Das Datenmodell wird zur Konsultation und zum Testen durch die Community veröffentlicht. ===== Schemas und Schlüsselkonzepte ===== Tabelle: https://tib.cloud/s/ZKNAAo3B8ATXsAP * Schema * Terminologiedienst * Kontrolliertes Vokabular * Taxonomie * Ontologie * Wissensgraph Tabelle X: Link: https://tib.cloud/s/ZKNAAo3B8ATXsAP In Linked Open Data (LOD) verwendete Terminologie DE {| class="wikitable" ! Konzept ! Wikidata-Link (Konzept) ! Hauptschwerpunkt ! Analogie ! Beispielressource ! URL ! Anwendungsbeispiel ! URL |- | Schema | Q1397073 | Datenstruktur | Die Vorlage. Konzeptionelles Schema / Datenmodell | Schema.org | [https://schema.org/] | VisualArtwork | [https://schema.org/VisualArtwork] |- | | | | | | | Smithsonian American Art Museum (SAAM) „In der Sierra Nevada, Kalifornien“ | [https://www.wikidata.org/wiki/Q20475372] |- | Terminologiedienst | Q22692845 | Verbreitung | Eine Bibliothek mit Vokabularen, Schemata, Ontologien usw. | TIB-Terminologiedienst | [https://terminology.tib.eu/ts/] | NFDI4CULTURE | [https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE] |- | Kontrolliertes Vokabular | Q1469824 | Konsistenz | Das Wörterbuch | Integrierte Normdatei / die Gemeinsame Normdatei (GND) | [https://portal.dnb.de/opac/showShortList] | Personen: Dürer, Albrecht | [https://d-nb.info/gnd/117751669] |- | Taxonomie | Q8269924 | Hierarchie | Sortierung nach Typ (allgemeine Klassifizierung) | Getty Art & Architecture Thesaurus (AAT) | [https://www.getty.edu/research/tools/vocabularies/aat/] | Deutscher Surrealist Max Ernst (verwendete Maltechniken) |[https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20und%20Grattage,in%20seinen%20Zeichnungen%20von%201925]. |- | | | | | Iconclass | [https://iconclass.org/] | Max Ernsts „Die Jungfrau, die das Christkind versohlt“ (Parady) | [https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926] |- | Ontologie | Q324254 | Semantik: Bedeutung & Logik (Informationswissenschaft) | Das Regelwerk oder der Stilführer | CIDOC (Comité International pour la DOCumentation / Internationales Komitee für Dokumentation) | [https://cidoc-crm.org/] | Sloane Lab Knowledge Base – Zusammenführung von 3 Sammlungen | [https://knowledgebase.sloanelab.org/resource/Start] |- | Wissensgraph | Q33002955 | Netzwerk von Dingen und Beziehungen | Eine Navigationskarte | Verzeichnis antiker Kunstwerke und architektonischer Bauwerke, die in der Renaissance bekannt waren | [https://www.census.de/] | Artemis-Suche | [https://database.census.de/#/detail/10013099] |- | | | | | Forschungsbereich | [https://researchspace.org/] | Hokusai: Das große Bilderbuch von allem |[https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start] |} EN {| class="wikitable" |- ! **Concept** ! **Wikidata link (Concept)** ! **Primary Focus** ! **Analogy** ! **Example resource** ! **URL** ! **Example use** ! **URL** |- | Schema | Q1397073 | Data Structure | The Template. Conceptual schema / data model | Schema.org | https://schema.org/ | VisualArtwork | https://schema.org/VisualArtwork |- | | | | | | | Smithsonian American Art Museum (SAAM) "Among the Sierra Nevada, California" | https://www.wikidata.org/wiki/Q20475372 |- | Terminology Service | Q22692845 | Distribution | A Library of Vocabularies, Schemas, Ontologies, etc | TIB Terminology Service | https://terminology.tib.eu/ts/ | NFDI4CULTURE | https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE |- | Controlled Vocabulary | Q1469824 | Consistency | The Dictionary | Integrated Authority File / die Gemeinsame Normdatei (GND) | https://portal.dnb.de/opac/showShortList | Persons: Dürer, Albrecht | https://d-nb.info/gnd/117751669 |- | Taxonomy | Q8269924 | Hierarchy | Sorting things by type (general classification) | Getty Art & Architecture Thesaurus (AAT) | https://www.getty.edu/research/tools/vocabularies/aat/ | German Surrealist Max Ernst (painting techniques used) | https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20and%20Grattage,in%20his%20drawings%20in%201925. |- | | | | | Iconclass | https://iconclass.org/ | Max Ernst’s "The Virgin Spanking the Christ Child" (Parady) | https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926 |- | Ontology | Q324254 | Semantics: Meaning & logic (information science) | The Rulebook or Writing Style Guide | CIDOC (Comité International pour la DOCumentation / International Committee for Documentation) | https://cidoc-crm.org/ | Sloane Lab Knowledge Base - unifying 3 collections | https://knowledgebase.sloanelab.org/resource/Start |- | Knowledge Graph | Q33002955 | Network of things and relations | A Navigational Map | Census of Antique Works of Art and Architecture Known in the Renaissance | https://www.census.de/ | Artemis search | https://database.census.de/#/detail/10013099 |- | | | | | Research Space | https://researchspace.org/ | Hokusai: The Great Picture Book of Everything | https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start |} ===== Schema-Übung ===== Zu bearbeitende Tabelle: https://tib.cloud/s/PicTdwCEqCQ6pBp (Passwort: bim2026) Wir werden uns mit folgenden Themen befassen: Ausstellung, Künstler und Katalog. '''''Geben Sie die gefundenen URLs ein. Fügen Sie bei Bedarf neue Zeilen, Spalten und Kommentare hinzu. Führen Sie sowohl manuelle als auch KI-Suchen durch, um die Ergebnisse zu vergleichen.''''' ===== Übung Nr. 1: Trage Links zu passenden Elementen aus den folgenden Quellen in die Tabelle ein: ===== * Wikidata:WikiProject Exhibitions/Properties * Generisches Wikibase-Modell für Kulturdaten – Wikibase4Research NFDI4Culture * CIDOC CRM (vollständig) * Terminologiedienst (NFDII4Culture) * Wikidata ===== Übung Nr. 2: Verwenden Sie KI-LLM, um passende Elemente zu finden ===== * <nowiki>https://gemini.google.com/</nowiki> ==== Prototyping ==== Entweder in dieser oder in der nächsten Sitzung wird die Gruppe in Teams aufgeteilt. ===== Schema ===== # Entwicklung eines Datenmodells: „Objekte in einer Ausstellung“ ===== Teile der Quarto-Publikation ===== # Ein Katalog einer Ausstellung des Sprengel Museums # Ein Katalog aller Ausstellungen und Ausstellungskataloge # Katalog der Ausstellungsbeiträge # Ein Glossar mit Begriffen – Personen und benannte Entitäten – aus Wikidata --- ===== Einführung in Quarto und Einfügen eines Ausstellungsbeitrags ===== Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace, Copilot: Agentic Coding) '''Voraussetzungen''' # Ein Laptop oder Computer, auf dem Sie VScode installieren können # Sie benötigen 2FA auf Ihrem Mobilgerät # Erstellen Sie ein GitHub-Konto # Installiere VScode # Verbinden Sie Ihr GitHub-Konto mit VScode # Erstellen Sie ein GitHub-Repository '''Klonen:''' https://github.com/mrchristian/prototype '''Modell: Auto''' '''So wurde das Repo eingerichtet. Agent-Eingabeaufforderungen:'''<blockquote>Ich möchte ein Quarto-Website-Projekt ausführen, bitte richte die Grundlagen ein. Das Projekt wird auf GitHub Pages veröffentlicht. Lege das Ausgabeverzeichnis auf „docs“ fest.</blockquote>Erstellen Sie eine Seite für das Quarto-Projekt, die die für diesen Wikidata-Eintrag verwendeten Daten abruft und als professionelle Webseite rendert <Fügen Sie hier Ihre Ausstellung ein – oder verwenden Sie diese> https://www.wikidata.org/wiki/Q138547468 Der Ansatz sollte eine SPARQL-Abfrage für die Daten erstellen und diese dann mithilfe eines Jupyter-Notebooks als HTML rendern. Alle Einträge: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Aufgaben ===== * Ausstellung ändern * Notebook ausführen * Quarto ausführen und Vorschau anzeigen * Auf Ihren GitHub Pages veröffentlichen ===== Schritt für Schritt ===== '''Teil 1: Arbeitsumgebung''' '''''HINWEIS: Sollten bei der lokalen Ausführung Probleme auftreten, nutzen Sie bitte die Online-Option von Codespace.''''' # Erstelle ein GitHub-Konto – https://github.com/ # Richte die Zwei-Faktor-Authentifizierung (2FA) ein – in der Regel auf dem Handy (Google Authenticator) # Installiere VSCode – https://code.visualstudio.com/download # Installiere GitHub Desktop – https://desktop.github.com/download/ # Füge dein GitHub-Konto hinzu, wenn du dazu aufgefordert wirst, und verwende die 2FA Schritt 2: Der Prototyp # Forken Sie das Repository: https://github.com/mrchristian/prototype # Wenn Sie lokal arbeiten, fahren Sie fort – wenn Sie Codespace verwenden, starten Sie Codespace (siehe unten und fahren Sie dann fort) # Testen Sie Quarto im Terminal: ## quarto check ## quarto render ## quarto preview (Strg+C – zum Beenden) # Falls es nicht funktioniert, führen Sie Quarto über den Agent aus # Ändern Sie die Wikidata-Ausstellung im Notebook # Notebook ausführen # quarto render und quarto preview ausführen # Alles speichern # Git: Nachricht, Commit und Push # Auf GitHub.com dein Repository ## Seiten aktivieren: GitHub Actions ## Code: Über das Zahnrad – Klicke auf „Meine GitHub Pages verwenden“ ## Registerkarte „Actions“: Quarto-Projekt veröffentlichen # ENDE – Wiederholen :-) ===== Codespace-Option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace ist eine virtuelle Maschine, die über GitHub gestartet werden kann. Das Repository enthält eine Dev-Container-Konfiguration, sodass du vollständig im Browser arbeiten kannst, ohne etwas lokal installieren zu müssen. 1.    Klicke auf der Repository-Seite auf GitHub auf „Code“ → „Codespaces“ → „Codespace erstellen“ auf der Hauptseite. 2.    Warte, bis der Container erstellt ist – Python-Pakete aus der Datei „requirements.txt“ werden automatisch installiert – dies dauert etwa 5 Minuten. 3.    Sobald alles installiert ist, kann der Codespace jederzeit genutzt werden. Er fährt automatisch herunter, wenn er nicht genutzt wird, und kann jederzeit neu gestartet werden. 4.    In Codespace geleistete Arbeit muss zurück ins Repository gepusht werden. 5.    Wenn Codespace 28 Tage lang nicht genutzt wird, wird der Codespace gelöscht. --- ===== Hausaufgabe – Sitzung Nr. 4 ===== * Hol alle Bücher aus der HsH-Bibliothek, die Ausstellungskataloge des Sprengel Museums sind. Bring sie zur nächsten Vorlesung mit * Erstelle einen Ausstellungs-Eintrag, falls noch nicht geschehen * Arbeite mit VSCode und dem Agent und experimentiere ==== Sitzung 5: Prototypenerstellung: Running Quarto Prototype, Federation, Prototype Teams ==== * Running Quarto Prototype - wie oben - https://github.com/mrchristian/prototype * DNB data download https://github.com/NFDI4Culture/linked-open-exhibition * Data Federation - WB4R ===== Links ===== https://wikibase.wbworkshop.tibwiki.io/wiki/Main_Page Glossar - https://nfdi4culture.github.io/linked-open-exhibition/Documentation/glossary ===== DNB Suche ===== https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books Sprengel Museum, 602 Artikel '''Über die Dienstleistungen von DNB''' https://www.dnb.de/librarylab https://deutsche-nationalbibliothek.github.io/jupyterlite/lab/ '''AUCH''' https://wiki.dnb.de/spaces/LINKEDDATASERVICE/pages/449878933/DNB+SPARQL+Service+BETA ===== Prototype Teams ===== * DNB-Daten * Katalog durchsuchen * Ausstellungsbeiträge * Vollständiges Datenmodell (alle) ENDE --- == EN == ''Materials and Tasks for the module "BIM-126-02, SoSe 2026, Worthington/Blümel" for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.'' Project GitHub repo: https://github.com/NFDI4Culture/linked-open-exhibition ==== Summary ==== The eight session course covers an introduction to Linked Open Data (LOD) in the context of : # Open Galleries Libraries Archives and Museums (GLAM), and; # The use of Wikimedia Foundation platforms. The Wikimedia Foundation platforms that will be used are: Wikidata; Wikibase, MediaWiki, and Wikimedia Commons. AI LLM will be used in the workflows: Code assistant ''copilot'', and a variety of AI LLM chat services for file generation and configurations to create SPARQL queries, Jinja 2.0 templates, etc. „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat The Methodologies employed are: Open-source software, Open Science, and rapid prototyping. ==== Linked Open Exhibition ==== The question being explored for the class is how can LOD be uséd to benefit museum exhibitions as Linked Open Exhibitions – a record of the exhibition, a catalogues of items in an exhibition, and other important data? As examples '''to gain exhibitions increased visitors numbers and create greater depth of engagement'''. With a focus of the question on how to make LOD records of '''items in an exhibition'''. ==== Learning points – In order of priority ==== # '''Wikidata/Wikibase LOD concepts:''' Items, Properties, Values, Qualifiers, Wikibase schemas, Classes, Lexemes, Knowledge Base, and Knowledge Graphs. # '''Linked Open Data (LOD):''' Semantic web, 5 star, RDF/Triples, Ontologies, Taxonomies, and controlled vocabularies. # '''Using LOD source:''' Identifiers, PIDs, information sources, media sources, and import and export tooling. # '''Data modelling:''' Methodologies, schema use, visualisation, and testing. # '''Data workflow tools:''' Git, IDE, AI code assistant (copilot), AI Chat, using Wikimedia Foundation tooling, data import and export tools, generating PIDs and making deposits in a scholarly repository. # '''Data presentation and data use:''' Wikidata Query Service results, MediaWiki infoboxes, AI Chat SPARQL query processing. # '''Open Science practice:''' Open-source software, Open Notebook Science, Open Licencing, PIDs, FAIR Data Principles, and ethical and good practice AI use. ==== Sessions ==== The sessions would be about cataloguing Sprengel Museum exhibitions using LOD and how to make visualisations and presentations. '''Learning to use LOD is the goal of the learning.''' The method will be to build out from a kernel of an ‘exhibition’ and add ‘item in an exhibition’. From the start the students will be the ones who make the LOD. This will start with minimal entries my by the students, then layering these up with – Identifiers, LOD Media sources, schemas, etc. And finally moving onto how to present the data in a way that satisfies the ‘use case’: '''to gain exhibitions increased visitors numbers and create greater depth of engagement'''. Here presentation technologies are used: MediaWiki infoboxes, Wikidata Query Service results, AI Chat SPARQL queries and other features, etc. ===== Session 1: Exhibition timeline creation - build out, add exhibitions ===== # Record minimal information for an exhibition in Wikidata as Linked Open Data: Title, museum, date, etc. e.g., https://www.wikidata.org/wiki/Q138547468 – See: Table 1: ''Minimal data entries for an exhibition'' # View the exhibition record in Wikidata Query Service results link (timeline and graph https://w.wiki/J8NJ | https://w.wiki/J8aS ) # Review exhibition entries. # Cover topics raised by making a LOD entry: Wikidata basics, Wikidata good practice, consulting schemas, importance of review and using GitHub Issues, comparing available data – before and after. ===== Session 2: Exhibition cataloguing - build up, add items, artists, catalogues ===== ===== Session 3: Museum visit - Sprengel Museum ===== ===== Session 4: Schemas and Prototyping (the end of class project) ===== ===== Session 5: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 6: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 7: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 8: Prototype Creation: Data entry, visualisation, and presentation ===== --- ==== Session 1: Exhibition timeline creation - build out, add exhibitions ==== The exercise: Create a Linked Open Data record for an exhibition using Wikidata (minimal entry). A. '''Creating the exhibition entry in Wikidata.''' # Login to Wikidata: https://www.wikidata.org/ # Have a source at hand to make a data entry, e.g., #* https://www.sprengel-museum.de/ausstellungen/archiv #* https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised #* https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&currentPosition=1 # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] ''Table'' ''1: Minimal data entries for an exhibition'' {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} '''''End of Session 1.''''' ==== Homework exercises ==== # Complete your allocated exhibition. Make sure all fields are complete from Table 1. If something cannot be added, either: A. Make a note in the exhibition allocation spreadsheet, or B. Send and email to [mailto:Simon.worththington@tib.eu simon.worththington@tib.eu] and I will help resolve your issue. '''Note: If you did not create an exhibition entry during the class make sure one is complete before the next class.''' # Create a GitHub account and add your GitHub handle next to your name, column ‘GitHub handle’, in the exhibition allocation spreadsheet. # Review your classmates exhibition entries. You have all been allocated a entry to review, see the Exhibition Allocation spreadsheet. Your name will be in column G. This first review has three questions – tick the boxes to show if each item has been complete and either add comments or correct the Wikidata exhibition entry. '''Note: If your allocated Exhibition entry hasn’t been made by you classmate then please contact them and ask them to complete the entry.''' Questions are: ## Are all the required fields present? ## Are all the fields correct? ## Is there an Deutsch and English entry? --- ==== Session 2: Exhibition cataloguing - build up, add items, artists, catalogues ==== The session has five exercies: # Exhibition update # Artist # Exhibition catalogue # AI LLM SPARQL experiments # <s>Artwork</s> The exercises include the following concepts: ==== Exercises ==== ==== 1. Exhibition updates ==== * Homework review: Complete all fields for an exhibition. Review your assigned review exhibition answering the three questions: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch</blockquote> * For the label. Convert words in all caps to sentence case. Use: https://convertcase.net/title-case-converter/ | Change from, e.g., ADRIAN SAUER: TRUTH TABLESPECTRUM INTERNATIONALER PREIS FÜR FOTOGRAFIE DER STIFTUNG NIEDERSACHSEN to Adrian Sauer: Truth Tablespectrum Internationaler Preis Für Fotografie Der Stiftung Niedersachsen. * Add the English language versions. Use DeepL to translate: https://www.deepl.com/en/translator ** Title: Add English title * Add the following. Change P710 Teilnehmer (Participant) to P921 zentrales Thema '''artists name.''' ** Qualifier on central theme to indicate the person is contributing artwork. * Use: Qualifier P170 creator and add artist Q483501 (type artists and it will automcomplete) * Reference: Gemeinsame Normdatei (GND) ID for a person, e.g., Gabriela Jolowicz https://d-nb.info/gnd/134184963 | Search your persons name and copy in the last part of number 134184963 * Talk page: Add in the review questions for your Wikidata entry: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch?</blockquote>Notice the useful links that tell you more about connected Linked Open Data! Note: SPARQL query showing data model. Properties and and values. Results: https://w.wiki/JMLX Made with Gemini AI: https://gemini.google.com/share/c43f34a67f67 ==== Concepts ==== * Wikidata parts – see about and diagram: ** https://www.wikidata.org/wiki/Wikidata:Introduction/de ** https://www.wikidata.org/wiki/Wikidata:Introduction#/media/File:Datamodel_in_Wikidata.svg * Applying a review process using Talk pages * Adding References * Using a type of LOD source – '''An authority record''' Gemeinsame Normdatei (GND) ID https://portal.dnb.de/opac.htm * SPARQL query --- ==== 2. Artists ==== The objective here is to ensure all artists have been included in exhibition listing and to then review the existing artists entry. Later a SPARQL query will be made to compare statements about all the artists in our dataset. * Before reviewing artists items make sure all artists have been listed in the exhibition item, with qualifier of being an artist and a reference to their GND record. ===== Important statements ===== {| class="wikitable" |Concept |CIDOC CRM (Full) |Linked Art (Selection) |Wikidata Equivalent |Note |- |Entity |E21 Person |Person |Q5 (human) |The base instance. |- |Label/Name |P1 is identified by → E33_E41 |identified_by (Name) | |Linked Art flattens this into a simple list of names. |- | | | |P735 Given name | |- | | | |P734 Family name | |- |Profession |P2 has type → E55 Type |classified_as |P106 (occupation) |Map to AAT 300025103 (artist). |- |Birth |P98i was born → E67 Birth |born (Birth) |P569 (date of birth) |CRM treats birth as an event; Wikidata as a property. |- |Death |P100i died in → E69 Death |died (Death) |P570 (date of death) |If the artist is still living, this is omitted. |- |Nationality |P107i member of → E74 Group |classified_as (Type) |P27 (citizenship) |Linked Art often models nationality as a Type. |- |Reference |P1 identifies ← E42 Identifier |identified_by (Identifier) |QID (The URI itself) |Used to link to external authorities (ULAN, VIAF). |- |Commons category |? |? |P373 search name |<nowiki>https://commons.wikimedia.org/</nowiki> |} From Google Gemini: https://gemini.google.com/share/578cc1b886d0 --- ===== Schemas and communities need consulting. ===== From Wikimedia: * WikiProject Visual Arts: https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Visual_arts * Wikiproject Exhibitions: https://www.wikidata.org/wiki/Wikidata:WikiProject_Exhibitions Semi-formal Generic Wikibase Model for Cultural Data: https://kgi4nfdi.github.io/Guidelines/guide/wikibase/data_modelling_import/ Formal: CIDOC Conceptual Reference Model (CRM) - https://cidoc-crm.org/ Linked Art (based on CIDOC) https://linked.art/model/actor/ ==== Concepts ==== * Data modeling * Schemas * Use case * Bottom up design * Identifiers --- ==== 3. Exhibition Catalogue ==== Search in both of these two places to find information about the catalogue for your assigned exhibition. * Sprengel Museum publication catalogue - https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised * DND (example) you can search for the exhibition name or Sprengel Museum '''-''' https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true ''Note: Make a note of any links you find in the exhibition listings spreadsheet.'' ===== Make a Wikidata entry for the catalogue ===== Note: first search for publication before making Wikidata entry. Use title, use ISBN, use GND. An example publication from DNB and Sprengel Shop. * https://portal.dnb.de/opac/showFullRecord?currentResultId=Gabriela+and+Jolowicz%26any&currentPosition=0 * https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised ===== Enter these statements ===== Note: Remember Label and Description {| class="wikitable" |Property |Label |Description/Example |- |P31 |instance of |catalogue (Q2352616) |- |P1476 |title |The official title of the catalogue (e.g., Vermeer and the Masters of Genre Painting) |- |P50 |author |The main curator or art historian (item link) |- |P123 |publisher |The museum or publishing house (e.g., Louvre Museum) |- |P577 |publication date |Year of release (e.g., 2024) |- |P212 |ISBN-13 |The 13-digit standard book identifier |- | |GND |ID |- |P973 |described at URL |A link to the catalogue's page on the museum’s website |} Google Gemini https://gemini.google.com/share/9a21f5522192 Example input: https://www.wikidata.org/wiki/Q138646145 ===== Link the record back to the exhibition ===== P972 Title ==== Concepts ==== * Data modeling * Identifier * Data as CC Zero / Copyright of data --- ==== 4. AI LLM SPARQL experiments ==== The Wikidata has a SPARQL interface where the LOD in Wikidata can be searched (queried) and outputted in a number of ways, formats, and a visualisations. As well as being saved on the web. We will us AI LLM chat to generate SPARQL queries. Later we will learn the fundamentals of writing a SPARQL query. But for the moment we want to see how they have be generated, the options, and creative applications. Using chat services or code assistants can be a valuable way to learn about new technologies. {| class="wikitable" |Service |Best For |Standout Feature |Key Model(s) |- |'''ChatGPT''' |General Use & Tasks |Deep Research & Agent Mode |GPT-5.4, GPT-5 |- |'''Claude''' |Coding & Writing |Artifacts (interactive workspace) |Claude 4.5, 4.6 |- |'''Google Gemini''' |Google Ecosystem |Nano Banana (native image/video) |Gemini 3.1 Pro |- |Perplexity |Real-time Research |Native Citations & Search Labs |Sonar, GPT-5, Claude |- |MS Copilot |Office Productivity |Copilot Vision & 365 Integration |GPT-5.2, Prometheus |- |DeepSeek |Logical Reasoning |High-tier performance at low cost |DeepSeek-V3, R1 |- |Grok |Real-time Social Info |Unfiltered X (Twitter) integration |Grok 4.1 |- |'''Meta AI''' |Social Media |Seamless integration in WhatsApp/IG |Llama 4 (Scout) |- |Poe |Model Testing |Access multiple LLMs in one app |Multi-model aggregator |- |Mistral (Le Chat) |Privacy & Developers |European-hosted, GDPR-focused |Mistral Large 3 |} Some of these can also be used via KISSKI „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat  === The exercise === The group will be split into a number of Zoom breakout groups and then the group spends 20 minutes experimenting generating SPARQL queries and other creative applications. Paste in results here: https://tib.cloud/apps/files/files/8251374?dir=/NFDI4Culture/HsH/BIM26/bim26-shared&editing=false&openfile=true Each room is assigned a Chat engine. Maximum there will be four groups. ·      Group #1: '''ChatGPT''' ·      Group #2: '''Claude''' ·      Group #3: '''Google Gemini''' ·       Group #4: '''Meta AI''' === Example exercise === Chat bots can read  a SPARQL query or a Wikidata address. e.g., Item https://www.wikidata.org/wiki/Q138547468 query graph https://w.wiki/JPNc query timeline https://w.wiki/JPPN Item Sprengel Museum https://www.wikidata.org/wiki/Q510144 Then the chatbot can be instructed to do things based on the information provided. You should ask the chat bot to generate Wikidata SPARQL queries and then paste the queries into the SPARQL querie interface. https://query.wikidata.org/ Use these examples and invent your own: # Create dashboard (count of things) # Create inventory (table) # Create graph data model Some output SPARQL queries ·       Map of artists place of birth - https://w.wiki/JPT3 ·       List of exhibitions - https://w.wiki/JPR3 ·       As plot of exhibitions - https://w.wiki/J8aS ==== Homework: Session 2 ==== Create a bottom up data model of an artwork in an exhibition. Include only the minimum information needed. The result should be a table like the ones presented for exhibition, artist, and catalogue. The table should include properties and attributes. You should consult the schemas mentioned above. You can use AI but attribute the AI and link to your question. If you use AI review the results and make notes about what you changed. Note: Think about how parts are related and what you need to add and what already exists in Wikidata. Submit your results as a spreadsheet or table. ===== Session 3: Museum visit - Sprengel Museum ===== 19 March 2026 ===== Session #4: Schemas and Prototyping (the end of class project) ===== ===== Recap and outline ===== Done * Creating exhibition entries in Wikidata * Filling our data models for Artist and Catalogue * Exploring the museum and its activities to help steer the prototype To do * Decide on the ideas for the prototype * Data model for items in an exhibition (Artwork and Exhibition) * Complete a data model for the end of the project that can be used by museums and complies to the sector standards – CIDOC and Wikidata. ===== What have we learned about the ‘Museum’s Story’ ===== TBC ===== Schemas ===== An opportunity to become familiar with how Linked Open Data is structured using common agreements on working practices. Over the period of the course a data model will be developed and finalised to describe ‘items in an exhibition’. The data model will be published for community consultation and testing. ===== Schemas and key concepts ===== Table: https://tib.cloud/s/ZKNAAo3B8ATXsAP * Schema * Terminology Service * Controlled Vocabulary * Taxonomy * Ontology * Knowledge Graph Table X: link: https://tib.cloud/s/ZKNAAo3B8ATXsAP Terminology used in Linked Open Data (LOD) {| class="wikitable" |- ! **Concept** ! **Wikidata link (Concept)** ! **Primary Focus** ! **Analogy** ! **Example resource** ! **URL** ! **Example use** ! **URL** |- | Schema | Q1397073 | Data Structure | The Template. Conceptual schema / data model | Schema.org | https://schema.org/ | VisualArtwork | https://schema.org/VisualArtwork |- | | | | | | | Smithsonian American Art Museum (SAAM) "Among the Sierra Nevada, California" | https://www.wikidata.org/wiki/Q20475372 |- | Terminology Service | Q22692845 | Distribution | A Library of Vocabularies, Schemas, Ontologies, etc | TIB Terminology Service | https://terminology.tib.eu/ts/ | NFDI4CULTURE | https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE |- | Controlled Vocabulary | Q1469824 | Consistency | The Dictionary | Integrated Authority File / die Gemeinsame Normdatei (GND) | https://portal.dnb.de/opac/showShortList | Persons: Dürer, Albrecht | https://d-nb.info/gnd/117751669 |- | Taxonomy | Q8269924 | Hierarchy | Sorting things by type (general classification) | Getty Art & Architecture Thesaurus (AAT) | https://www.getty.edu/research/tools/vocabularies/aat/ | German Surrealist Max Ernst (painting techniques used) | https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20and%20Grattage,in%20his%20drawings%20in%201925. |- | | | | | Iconclass | https://iconclass.org/ | Max Ernst’s "The Virgin Spanking the Christ Child" (Parady) | https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926 |- | Ontology | Q324254 | Semantics: Meaning & logic (information science) | The Rulebook or Writing Style Guide | CIDOC (Comité International pour la DOCumentation / International Committee for Documentation) | https://cidoc-crm.org/ | Sloane Lab Knowledge Base - unifying 3 collections | https://knowledgebase.sloanelab.org/resource/Start |- | Knowledge Graph | Q33002955 | Network of things and relations | A Navigational Map | Census of Antique Works of Art and Architecture Known in the Renaissance | https://www.census.de/ | Artemis search | https://database.census.de/#/detail/10013099 |- | | | | | Research Space | https://researchspace.org/ | Hokusai: The Great Picture Book of Everything | https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start |} ===== Schemas exercise ===== Spreadsheet to work on: https://tib.cloud/s/PicTdwCEqCQ6pBp (password: bim2026) We will be looking at: Exhibition, Artist, and Catalogue. '''''Enter the URLs found. Add new rows, columns, comments if needed. Keep manual searches as well as AI searches for comparison.''''' ===== Exercise #1: Enter links into the spreadsheet of matching items from the following: ===== * Wikidata:WikiProject Exhibitions/Properties * Generic Wikibase Model for Cultural Data - Wikibase4Research NFDI4Culture * CIDOC CRM (Full) * Terminology Service (NFDII4Culture) * Wikidata ===== Exercise #2: Use AI LLM to find matching items ===== * https://gemini.google.com/ ==== Prototyping ==== Either in this session or in the next session the group will be divided into teams. ===== Schema ===== # Data model development: ‘items in an exhibition’ ===== Quarto publication parts ===== # A catalogue of a Sprengel Museum exhibition # A catalogue of all exhibitions and exhibition catalogues # Catalogue of exhibition entries --- ==== Learning to use Quarto and inserting an exhibition entry ==== Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype '''Model: Auto''' '''How the repo was setup. Agent promts:'''<blockquote>''I want to run a Quarto website project, please setup the basics. The project will be published on GitHub Pages. Set the output directory to docs.'' </blockquote>Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minutes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- ===== Homework - session #4 ===== * Get all books from HsH library that are Sprengel Museum exhibition catalogues. Bring to the next class * Make an exhibition entry if not done * Work with VSCode and the Agent and experiment * Add entries from existing ontologies: https://tib.cloud/s/PicTdwCEqCQ6pBp?dir=/&editing=false&openfile=true ==== Sitzung 5: Prototyping: Running Quarto Prototype, Federation, Prototype Teams ==== * Running Quarto Prototype - https://github.com/mrchristian/prototype * DNB data download https://github.com/NFDI4Culture/linked-open-exhibition * Data Federation - WB4R ===== Links ===== https://wikibase.wbworkshop.tibwiki.io/wiki/Main_Page Glossar - https://nfdi4culture.github.io/linked-open-exhibition/Documentation/glossary ===== DNB Search ===== https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books Sprengel Museum, 602 Artikel '''About the DNB''' https://www.dnb.de/librarylab https://deutsche-nationalbibliothek.github.io/jupyterlite/lab/ '''Also''' https://wiki.dnb.de/spaces/LINKEDDATASERVICE/pages/449878933/DNB+SPARQL+Service+BETA ===== Prototype Teams ===== * DNB Data * Katalog scan * Exhibition entries * Datenmodell (alle) ===== --- ===== == Session 6: Class Project – Prototyping: ''Linked Open Exhibitions'' == Prototype URL (currently 2026-04-29 a shell framework): https://nfdi4culture.github.io/linked-open-exhibition/ ==== Program: ==== 11:30 – 11:50 (20 min) '''Outline: Class Project – Prototyping: Linked Open Exhibitions.''' What is it and what needs to be delivered. Allocation to sub-project and tasks outline. '''Activity #1: Bottom-up data modeling: Data mapping''' 11:50 – 12:20 (30 min) Data finding and exploration (break out rooms) 12:20 – 12:40 (20 min) Review data findings (class discussion) 12:40 – 12:55 (15 min) Pause Break '''Activity #2: Top-down data modeling: Schema mapping''' 12:55 – 13:35 (40 min) Map data against schemas (break out rooms) 13:35 – 13:55 (20 min) Review findings (class discussion) '''13:55 – 14:15 (20 min) Work time: Open time-slot to review running ‘Tech Stack’ or address any other questions''' --- ==== Important links ==== * '''Main Prototype repository:''' https://nfdi4culture.github.io/linked-open-exhibition/ * Quarto setup: ‘''BIM Prototype 02 Quarto Website’:'' https://mrchristian.github.io/prototype/ * Instructions for ‘Tech Stack’: ''Einführung in Quarto und Einfügen eines Ausstellungsbeitrags'' [[BIM-126-02-Data-Science-Linked-Open-Exhibition#Einführung in Quarto und Einfügen eines Ausstellungsbeitrags|https://en.wikiversity.org/wiki/BIM-126-02-Data-Science-Linked-Open-Exhibition#Einf%C3%BChrung_in_Quarto_und_Einf%C3%BCgen_eines_Ausstellungsbeitrags]] * Earlier Prototype (2025): https://nfdi4culture.github.io/open-museum/ ==== ''Outline: Class Project – Prototyping: Linked Open Exhibitions'' ==== Prototype: https://nfdi4culture.github.io/linked-open-exhibition/ Repo: https://nfdi4culture.github.io/linked-open-exhibition/ Why? * Rapid Prototyping in this context is used to learn about ‘Data Modeling using Linked Open Data’. '''''NB: The data modeling skills and experience learned here is a core competence that gives a foundation to be able to create data models in a wide set of professional contexts.''''' ** How to do data modeling ** To use method: Bottom-up; KISS (Keep it Short and Simple); Top-down ** Evaluation and validation ** Operationalize a data model ** User testing ** Good practice, including Open Scholarship (Open Science) practice. e.g. FAIR Data Principles ** Experiment with AI LLMs and agentic coding in the workflow * Rapid Prototyping is a Design Research methodology – meaning to create or discover knowledge by doing. What? Create a website prototype a the whole class: https://nfdi4culture.github.io/linked-open-exhibition/ The website is made of three data driven sub-projects: # Manual Wikidata entries for Sprengel Museum website – class entries already made # Bulk exhibition entries derived from 600+ DNB records for ‘Sprengel Museum’ - imported # HsH Library information records for a search on the Sprengel Museum and one scan of a Sprengel Museum catalogue for Text and Data Mining (TDM) – to do How? Simon Worthington will act as Publication Manager. This involves running or guiding complex software parts. Copilot agentic coding will be used (experimented with) for some parts. Class is divided into three teams of the sub-projects: # Sprengel Museum exhibitions website; # DNB records ‘Sprengel Museum’ # Text and Data Mining: Library catalogue Sprengel Museum Each team carries out the same tasks for their parts to complete a round of data modeling: # Collect data – bottom-up method # Validate the data – top-down method # Presentation of data – Quarto website ‘[https://nfdi4culture.github.io/linked-open-exhibition/ Linked Open Exhibitions]’ Goal: Definition-of-done (DoD) ''NB: Developer speak'' * A documented data model (Table) with diagram (Mermaid, GraphVis, or Draw.io) * Mapping of data model to schemas (Table) * Idea for presentation of data for each sub-section in the prototype and implementation with Publication Manager assistance, e.g., for DNB a chronological list of exhibitions with images. * Documentation of AI LLM use as an assistant, attribution, and comments on good practice * Data provenance and good practice checklist completion * The final result ‘Class Project – Prototyping: Linked Open Exhibitions’ will be made as an institutional deposit with [https://zenodo.org/ Zenodo]. --- ==== Activity #1: Collecting data and bottom up data modeling ==== Confirm, create, or expand existing data models by looking at the source. Each project has a source: * Team #1: Sprengel Museum website, exhibition listings: ** https://www.sprengel-museum.de/ and ** spreadsheet of entered exhibitions CSV GitHub | Spreadsheet TIB Cloud (passworded) ** In prototype: https://nfdi4culture.github.io/linked-open-exhibition/exhibitions.html * Team #2: DNB records of search for ‘Sprengel Museum’: ** https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books | https://wikibase.wbworkshop.tibwiki.io/ ** CSV https://github.com/NFDI4Culture/linked-open-exhibition/blob/main/catalogues/sprengel_exhibitions.csv ** Images of book covers https://github.com/NFDI4Culture/linked-open-exhibition/tree/main/catalogues/images * Team #3: HsH Library information on catalogues for ‘Sprengel Museum’: ** https://katalog.bib.hs-hannover.de/vufind/Search/Results?lookfor=Sprengel%2BMuseum ** Team 3 have to start from scratch as as yet we don’t have a library record or item in an exhibition data model. Tip: look back at the other models to start building up your data models. '''>>> Add to data model here:''' https://tib.cloud/s/PicTdwCEqCQ6pBp (passworded) ==== OBJECTIVE ==== To ensure that the data model can represent the source. Are there enough entries to describe the things that make up the source. The process is iterative, meaning it keeps on being repeated with improvements and changes being made. ==== TASKS ==== * Edit the purple grey area, thje green are will be edited in the next activity * Review and correct existing information * Add new concepts if the source needs it * Orange areas need filling in. The cells might need editing or added to. * Data types can be found on Property Pages only Items (QIDs) don’t have data types, in <nowiki>https://www.wikidata.org/wiki/Property:P1476</nowiki> under the Label - ''Data type'' * URI is equivelent to URL Tips * Look at other examples on Wikidata: Artists, exhibitions, catalogues, bibliographic, or items in an exhibition. * Use an AI to lookup schema explanations or options. Register with KISSKI to get better AI privacy use. ==== Activity #2: Top down data modeling validation ==== * Team #1: Sprengel Museum website * Team #2: DNB records of search for ‘Sprengel Museum’ * Team #3: HsH Library information on catalogues for ‘Sprengel Museum’ ===== OBJECTIVE ===== Map all concepts ==== TASKS ==== Look up the concept in the different resources and add mapping links, Work time: Open time-slot to review running ‘Tech Stack’ or address any other questions --- ===== Homework ===== * Complete the Bottom-up and Top-down modelling * Team #3: Visit the library and make a digital scan on a copier machine, store as PDF. The scan will be used for text and data mining and the file deleted and destroyed after. We will only be extracting metadata from the scan. * Come along to the next class with ideas and suggestions for what you would like to have displayed from your data and data models in the prototype. [[Category:Wikidata]] kyklexm2d7cymdeomyji4cjzurerf8b 2807040 2807036 2026-04-29T17:37:34Z Mrchristian 281704 2807040 wikitext text/x-wiki DE (EN Below) {{TOCleft}} ==== Linked Open Exhibition ==== ''Materialien und Aufgaben für das Modul „BIM-126-02, SoSe 2026, Worthington/Blümel” für Studierende der Hochschule Hannover. Die Materialien werden gemeinsam mit mehreren Kollegen aus dem [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab] der TIB Hannover erstellt.'' Projekt-GitHub-Repo: https://github.com/NFDI4Culture/linked-open-exhibition ==== Zusammenfassung ==== Der achtteilige Kurs bietet eine Einführung in Linked Open Data (LOD) im Kontext von: # Open Galleries Libraries Archives and Museums (GLAM) und # der Nutzung von Plattformen der Wikimedia Foundation. Die folgenden Plattformen der Wikimedia Foundation werden verwendet: Wikidata, Wikibase, MediaWiki und Wikimedia Commons. AI LLM wird in den folgenden Workflows verwendet: Code Assistant ''Copilot'' und eine Vielzahl von AI LLM-Chat-Diensten für die Dateierstellung und Konfigurationen zur Erstellung von SPARQL-Abfragen, Jinja 2.0-Vorlagen usw. Das „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) kann für unbegrenztes ChatGPT5 genutzt werden: https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat Die verwendeten Methoden sind: Open-Source-Software, Open Science und Rapid Prototyping. ==== Linked Open Exhibition ==== Die Frage, die in diesem Kurs untersucht wird, lautet: Wie kann LOD genutzt werden, um Museumsausstellungen als Linked Open Exhibitions zu verbessern – als Aufzeichnung der Ausstellung, als Katalog der Ausstellungsstücke und für andere wichtige Daten? Als Beispiele '''dienen die Steigerung der Besucherzahlen von Ausstellungen und die Schaffung einer größeren Tiefe des Engagements'''. Der Schwerpunkt liegt auf der Frage, wie LOD-Aufzeichnungen von '''Exponaten in einer Ausstellung''' erstellt werden können. ==== Lernpunkte – in der Reihenfolge ihrer Priorität ==== # '''Wikidata/Wikibase LOD-Konzepte:''' Objekte, Eigenschaften, Werte, Qualifikatoren, Wikibase-Schemas, Klassen, Lexeme, Wissensbasis und Wissensgraphen. # '''Linked Open Data (LOD):''' Semantic Web, 5-Sterne-Bewertung, RDF/Triples, Ontologien, Taxonomien und kontrollierte Vokabulare. # '''Verwendung von LOD-Quellen:''' Identifikatoren, PIDs, Informationsquellen, Medienquellen sowie Import- und Export-Tools. # '''Datenmodellierung:''' Methodiken, Schemaverwendung, Visualisierung und Testen. # '''Daten-Workflow-Tools:''' Git, IDE, KI-Code-Assistent (Copilot), KI-Chat, Verwendung von Wikimedia Foundation-Tools, Datenimport- und -export-Tools, Generierung von PIDs und Hinterlegung in einem wissenschaftlichen Repositorium. # '''Datenpräsentation und Datennutzung:''' Ergebnisse des Wikidata Query Service, MediaWiki-Infoboxen, Verarbeitung von SPARQL-Abfragen durch KI-Chat. # '''Open-Science-Praxis:''' Open-Source-Software, Open Notebook Science, Open Licensing, PIDs, FAIR-Datenprinzipien sowie ethische und bewährte Verfahren bei der Nutzung von KI. ==== Sitzungen ==== Die Sitzungen befassen sich mit der Katalogisierung von Ausstellungen des Sprengel Museums unter Verwendung von LOD und der Erstellung von Visualisierungen und Präsentationen. '''Das Ziel des Lernens ist es, den Umgang''' mit '''LOD''' zu '''erlernen.''' Die Methode besteht darin, ausgehend von einem Kern einer „Ausstellung” „Exponate in einer Ausstellung” hinzuzufügen. Von Anfang an sind es die Studierenden, die die LOD erstellen. Dies beginnt mit minimalen Einträgen der Studierenden, die dann mit Identifikatoren, LOD-Medienquellen, Schemata usw. ergänzt werden. Schließlich wird gezeigt, wie die Daten so präsentiert werden können, dass sie dem „Anwendungsfall” entsprechen: '''die Besucherzahlen der Ausstellungen zu steigern und ein tieferes Engagement zu erreichen'''. Hier kommen Präsentationstechnologien zum Einsatz: MediaWiki-Infoboxen, Ergebnisse des Wikidata Query Service, KI-Chat-SPARQL-Abfragen und andere Funktionen usw. ==== Sitzung 1: Erstellung einer Ausstellung-Zeitleiste – Aufbau, Hinzufügen von Ausstellungen ==== # Erfassen Sie minimale Informationen zu einer Ausstellung in Wikidata als Linked Open Data: Titel, Museum, Datum usw. Beispiel: https://www.wikidata.org/wiki/Q138547468 – Siehe: Tabelle1 : ''Minimale Dateneinträge für eine Ausstellung'' # Zeigen Sie den Ausstellungseintrag in Wikidata an Ergebnisse des Abfragedienstes anzeigen Link (Zeitleiste und Grafik https://w.wiki/J8NJ | https://w.wiki/J8aS ) # Überprüfen Sie die Ausstellungseinträge. # Behandeln Sie Themen, die durch die Erstellung eines LOD-Eintrags aufgeworfen werden: Wikidata-Grundlagen, bewährte Verfahren für Wikidata, Konsultation von Schemata, Bedeutung der Überprüfung und Verwendung von GitHub Issues, Vergleich der verfügbaren Daten – vorher und nachher. ==== Sitzung 2: Ausstellungskatalogisierung – Aufbau, Hinzufügen von Objekten, Künstlern, Katalogen ==== ==== Sitzung 3: Museumsbesuch – Sprengel Museum (noch zu bestätigen) ==== ==== Sitzung 4: Ausstellungskatalogisierung – Massenhinzufügungen: Hinzufügen von Objekten, Künstlern, Katalogen ==== ==== Sitzung 5: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 6: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 7: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== ==== Sitzung 8: Prototypenerstellung: Dateneingabe, Visualisierung und Präsentation ==== --- ==== Sitzung 1: Erstellung eines Ausstellungskalenders – Aufbau, Hinzufügen von Ausstellungen ==== Die Übung: Erstellen Sie einen Linked-Open-Data-Datensatz für eine Ausstellung mit Wikidata (Mindestangaben). A. '''Erstellen des Ausstellungseintrags in Wikidata.''' # Anmeldung bei Wikidata: https://www.wikidata.org/ # Halten Sie eine Quelle bereit, um Daten einzugeben, z. B. #* https://www.sprengel-museum.de/ausstellungen/archiv #* https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised #* https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&currentPosition=1 # Überprüfen Sie, ob es bereits einen Eintrag für die Ausstellung auf Wikidata gibt. Verwenden Sie dazu die Suchfunktion. # Erstellen Sie einen Eintrag oder bearbeiten Sie einen bestehenden Eintrag. #* Hinweis: Überprüfen Sie, welche Sprache Sie verwenden. Wir werden Einträge in Deutsch und Englisch hinzufügen (beginnend mit Deutsch). # Erstellen Sie die folgenden Dateneinträge in Wikidata, siehe: Tabelle 1: ''Minimale Dateneinträge für eine Ausstellung.'' # Überprüfen Sie die Wikidata-Einträge zur Ausstellung. Die Überprüfung erfolgt anhand von drei Fragen. Fügen Sie bei Bedarf Kommentare hinzu, Korrekturen können vorgenommen werden. Ergebnisse und Anmerkungen können auf der Diskussionsseite des Eintrags hinzugefügt werden, z. B. #* Alle Einträge vorhanden [ ] #* Alle Einträge sind korrekt [ ] #* Einträge sind in Deutsch und Englisch – im Rahmen des Zumutbaren [ ] ''Tabelle2 : Mindestdaten für einen Ausstellungseintrag'' {| class="wikitable" | colspan="7" |'''Felder, die zur Erstellung eines Ausstellungseintrags verwendet werden. Siehe Beispiel: https://www.wikidata.org/wiki/Q138547468''' |- |A |Beschriftung | colspan="5" |Hinweis: Kurz halten. Titel der Ausstellung verwenden |- |B |Beschreibung | colspan="5" |Hinweis: Zur Unterscheidung von anderen Einträgen verwenden. Folgen Sie diesem Beispiel: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Eigentum (P) und Objekt (Q)''' |'''URI''' |'''DE''' | |'''Hinzufügen''' |'''Anmerkung''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |Instanz von |Q464980 |Element hinzufügen |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Ausstellung | |(oben verwendet) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Titel |Titel |Klartext |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Standort |Sprengel Museum Hannover Q510144 |Artikel hinzufügen |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Startzeit |Datum |JJJJ-MM-TT |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |Endzeit |Datum |JJJJ-MM-TT |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Kurator |Person |Element hinzufügen (falls nicht vorhanden, muss erstellt werden/kann derzeit weggelassen werden) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Teilnehmer |Person (der Künstler) |Element hinzufügen (falls nicht vorhanden, muss erstellt werden/kann derzeit weggelassen werden) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |Offizielle Website |Offizielle Website |URL |URL |} Ende von Sitzung 1. ==== Hausaufgabenübungen ==== 1. Vervollständigen Sie Ihre zugewiesene Ausstellung. Stellen Sie sicher, dass alle Felder aus Tabelle 1 ausgefüllt sind. Wenn etwas nicht hinzugefügt werden kann, haben Sie zwei Möglichkeiten: A. Machen Sie eine Notiz in der Tabelle zur Ausstellungszuweisung oder B. Senden Sie eine E-Mail an [mailto:Simon.worththington@tib.eu simon.worththington@tib.eu] , damit ich Ihnen bei der Lösung Ihres Problems helfen kann. '''Hinweis: Wenn Sie während des Unterrichts keinen Ausstellungseintrag erstellt haben, stellen Sie sicher, dass dieser vor der nächsten Unterrichtsstunde fertiggestellt ist.''' 2. Erstellen Sie ein GitHub-Konto und fügen Sie Ihren GitHub-Namen neben Ihrem Namen in der Spalte „GitHub-Name” in der Tabelle zur Zuweisung der Ausstellungen hinzu. 3. Überprüfen Sie die Ausstellungseinträge Ihrer Klassenkameraden. Ihnen wurde allen ein Eintrag zur Überprüfung zugewiesen, siehe Tabelle zur Zuweisung der Ausstellungen. Ihr Name steht in Spalte G. Diese erste Überprüfung umfasst drei Fragen – kreuzen Sie die Kästchen an, um anzuzeigen, ob jeder Punkt ausgefüllt wurde, und fügen Sie entweder Kommentare hinzu oder korrigieren Sie den Wikidata-Ausstellungseintrag. '''Hinweis: Wenn der Ihnen zugewiesene Ausstellungseintrag nicht von Ihrem Klassenkameraden erstellt wurde, kontaktieren Sie ihn bitte und bitten Sie ihn, den Eintrag zu vervollständigen.''' Die Fragen lauten: 1. Sind alle erforderlichen Felder vorhanden? 2. Sind alle Felder korrekt ausgefüllt? 3. Gibt es einen deutschen und einen englischen Eintrag? --- === Sitzung 2: Ausstellungskatalogisierung – Aufbau, Hinzufügen von Objekten, Künstlern, Katalogen === ==== Die Sitzung umfasst fünf Übungen: ==== # Ausstellungsaktualisierung # Künstler # Ausstellungskatalog # AI LLM SPARQL-Experimente # <s>Kunstwerk</s> ==== Die Übungen umfassen die folgenden Konzepte: ==== ==== Übungen ==== ==== 1. Aktualisierung der Ausstellung ==== * Hausaufgabenüberprüfung: Füllen Sie alle Felder für eine Ausstellung aus. Überprüfen Sie die Ihnen zugewiesene Ausstellung, indem Sie die folgenden drei Fragen beantworten: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch</blockquote> * Für das Label. Wandeln Sie Wörter in Großbuchstaben in Satzschrift um. Verwenden Sie: https://convertcase.net/title-case-converter/ | Ändern Sie z. B. ADRIAN SAUER: TRUTH TABLESPECTRUM INTERNATIONALER PREIS FÜR FOTOGRAFIE DER STIFTUNG NIEDERSACHSEN in Adrian Sauer: Truth Tablespectrum Internationaler Preis Für Fotografie Der Stiftung Niedersachsen. * Fügen Sie die englischen Versionen hinzu. Verwenden Sie DeepL zum Übersetzen: https://www.deepl.com/en/translator ** Titel: Fügen Sie den englischen Titel hinzu * Fügen Sie Folgendes hinzu. Ändern Sie P710 Teilnehmer (Participant) in P921 zentrales Thema artists name. ** Qualifier zum zentralen Thema, um anzugeben, dass die Person Kunstwerke beisteuert. * Verwenden Sie: Qualifier P170 creator und fügen Sie artist Q483501 hinzu (geben Sie „Künstler” ein, es wird automatisch vervollständigt) * Referenz: Gemeinsame Normdatei (GND) ID für eine Person, z. B. Gabriela Jolowicz https://d-nb.info/gnd/134184963 | Suchen Sie den Namen der Person und kopieren Sie den letzten Teil der Nummer 134184963 * Diskussionsseite: Fügen Sie die Überprüfungsfragen für Ihren Wikidata-Eintrag hinzu: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch?</blockquote>Beachten Sie die nützlichen Links, die Ihnen mehr über verbundene Linked Open Data verraten! Hinweis: SPARQL-Abfrage zur Anzeige des Datenmodells. Eigenschaften und Werte. Ergebnisse: https://w.wiki/JMLX Erstellt mit Gemini AI: https://gemini.google.com/share/c43f34a67f67 ==== Konzepte ==== * Wikidata-Teile – siehe Informationen und Diagramm: ** https://www.wikidata.org/wiki/Wikidata:Introduction/de ** https://www.wikidata.org/wiki/Wikidata:Introduction#/media/File:Datamodel_in_Wikidata.svg * Anwendung eines Überprüfungsprozesses mithilfe von Diskussionsseiten * Hinzufügen von Referenzen * Verwendung einer LOD-Quelle – Ein Normdatensatz Gemeinsame Normdatei (GND) ID <nowiki>https://portal.dnb.de/opac.htm</nowiki> * SPARQL-Abfrage --- ==== 2. Künstler ==== Das Ziel hierbei ist es, sicherzustellen, dass alle Künstler in die Ausstellungsliste aufgenommen wurden, und anschließend die bestehenden Künstlereinträge zu überprüfen. Später wird eine SPARQL-Abfrage durchgeführt, um Aussagen über alle Künstler in unserem Datensatz zu vergleichen. Bevor Sie die Künstereinträge überprüfen, stellen Sie sicher, dass alle Künstler im Ausstellungseintrag aufgeführt sind, mit dem Qualifikationsmerkmal „Künstler” und einem Verweis auf ihren GND-Datensatz. ==== Wichtige Aussagen ==== {| class="wikitable" |Concept |CIDOC CRM (Full) |Linked Art (Selection) |Wikidata Equivalent |Note |- |Entity |E21 Person |Person |Q5 (human) |The base instance. |- |Label/Name |P1 is identified by → E33_E41 |identified_by (Name) | |Linked Art flattens this into a simple list of names. |- | | | |P735 Given name | |- | | | |P734 Family name | |- |Profession |P2 has type → E55 Type |classified_as |P106 (occupation) |Map to AAT 300025103 (artist). |- |Birth |P98i was born → E67 Birth |born (Birth) |P569 (date of birth) |CRM treats birth as an event; Wikidata as a property. |- |Death |P100i died in → E69 Death |died (Death) |P570 (date of death) |If the artist is still living, this is omitted. |- |Nationality |P107i member of → E74 Group |classified_as (Type) |P27 (citizenship) |Linked Art often models nationality as a Type. |- |Reference |P1 identifies ← E42 Identifier |identified_by (Identifier) |QID (The URI itself) |Used to link to external authorities (ULAN, VIAF). |- |Commons category |? |? |P373 search name |<nowiki>https://commons.wikimedia.org/</nowiki> |} Aus Google Gemini: https://gemini.google.com/share/578cc1b886d0 --- ==== Schemas und Communities benötigen Beratung. ==== '''Aus Wikimedia:''' WikiProject Visual Arts: https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Visual_arts Wikiproject Exhibitions: https://www.wikidata.org/wiki/Wikidata:WikiProject_Exhibitions '''Halbformell''' Generisches Wikibase-Modell für Kulturdaten: https://kgi4nfdi.github.io/Guidelines/guide/wikibase/data_modelling_import/ '''Formell:''' CIDOC Conceptual Reference Model (CRM) – https://cidoc-crm.org/ Linked Art (basierend auf CIDOC) https://linked.art/model/actor/ ==== Konzepte ==== * Datenmodellierung * Schemas * Anwendungsfall * Bottom-up-Design * Identifikatoren --- ==== 3. Ausstellungskatalog ==== Suchen Sie an beiden Orten nach Informationen zum Katalog Ihrer zugewiesenen Ausstellung. Sprengel Museum Publikationskatalog – https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised DND (Beispiel) Sie können nach dem Namen der Ausstellung oder dem Sprengel Museum suchen – https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true Hinweis: Notieren Sie sich alle Links, die Sie in der Tabelle mit den Ausstellungslisten finden. ===== Erstellen Sie einen Wikidata-Eintrag für den Katalog. ===== Hinweis: Suchen Sie zunächst nach der Veröffentlichung, bevor Sie einen Wikidata-Eintrag erstellen. Verwenden Sie den Titel, die ISBN und die GND. Ein Beispiel für eine Veröffentlichung aus DNB und Sprengel Shop. * https://portal.dnb.de/opac/showFullRecord?currentResultId=Gabriela+and+Jolowicz%26any&currentPosition=0 * https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised ===== Geben Sie diese Angaben ein ===== Hinweis: Denken Sie an die Bezeichnung und Beschreibung {| class="wikitable" |Property |Label |Description/Example |- |P31 |instance of |catalogue (Q2352616) |- |P1476 |title |The official title of the catalogue (e.g., Vermeer and the Masters of Genre Painting) |- |P50 |author |The main curator or art historian (item link) |- |P123 |publisher |The museum or publishing house (e.g., Louvre Museum) |- |P577 |publication date |Year of release (e.g., 2024) |- |P212 |ISBN-13 |The 13-digit standard book identifier |- | |GND |ID |- |P973 |described at URL |A link to the catalogue's page on the museum’s website |} Google Gemini https://gemini.google.com/share/9a21f5522192 Beispiel für eine Eingabe: https://www.wikidata.org/wiki/Q138646145 ==== Verlinken Sie den Datensatz zurück zur Ausstellung. ==== P972 > Titel ==== Konzepte ==== * Datenmodellierung * Identifikator * Daten als CC Zero / Urheberrecht der Daten --- ==== 4. AI LLM SPARQL-Experimente ==== Wikidata verfügt über eine SPARQL-Schnittstelle, über die die LOD in Wikidata durchsucht (abgefragt) und auf verschiedene Arten, in verschiedenen Formaten und Visualisierungen ausgegeben werden kann. Außerdem kann sie im Web gespeichert werden. Wir werden den AI LLM-Chat verwenden, um SPARQL-Abfragen zu generieren. Später werden wir die Grundlagen des Schreibens einer SPARQL-Abfrage lernen. Aber zunächst wollen wir sehen, wie sie generiert werden, welche Optionen es gibt und wie sie kreativ eingesetzt werden können. Die Verwendung von Chat-Diensten oder Code-Assistenten kann eine wertvolle Möglichkeit sein, um neue Technologien kennenzulernen. {| class="wikitable" |Service |Best For |Standout Feature |Key Model(s) |- |'''ChatGPT''' |General Use & Tasks |Deep Research & Agent Mode |GPT-5.4, GPT-5 |- |'''Claude''' |Coding & Writing |Artifacts (interactive workspace) |Claude 4.5, 4.6 |- |'''Google Gemini''' |Google Ecosystem |Nano Banana (native image/video) |Gemini 3.1 Pro |- |Perplexity |Real-time Research |Native Citations & Search Labs |Sonar, GPT-5, Claude |- |MS Copilot |Office Productivity |Copilot Vision & 365 Integration |GPT-5.2, Prometheus |- |DeepSeek |Logical Reasoning |High-tier performance at low cost |DeepSeek-V3, R1 |- |Grok |Real-time Social Info |Unfiltered X (Twitter) integration |Grok 4.1 |- |'''Meta AI''' |Social Media |Seamless integration in WhatsApp/IG |Llama 4 (Scout) |- |Poe |Model Testing |Access multiple LLMs in one app |Multi-model aggregator |- |Mistral (Le Chat) |Privacy & Developers |European-hosted, GDPR-focused |Mistral Large 3 |} Einige davon können auch über KISSKI genutzt werden. Das „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) kann für unbegrenztes ChatGPT5 genutzt werden: https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat ==== Die Übung ==== Die Gruppe wird in mehrere Zoom-Breakout-Gruppen aufgeteilt und verbringt dann 20 Minuten damit, SPARQL-Abfragen und andere kreative Anwendungen zu generieren. Fügen Sie die Ergebnisse hier ein: https://tib.cloud/apps/files/files/8251374?dir=/NFDI4Culture/HsH/BIM26/bim26-shared&editing=false&openfile=true Jedem Raum wird eine Chat-Engine zugewiesen. Es gibt maximal vier Gruppen. · Gruppe Nr. 1: ChatGPT · Gruppe Nr. 2: Claude · Gruppe Nr. 3: Google Gemini · Gruppe Nr. 4: Meta AI ==== Beispielübung ==== Chatbots können eine SPARQL-Abfrage oder eine Wikidata-Adresse lesen. z. B. * Artikel https://www.wikidata.org/wiki/Q138547468 * Abfragegrafik https://w.wiki/JPNc * Abfragetidsachse https://w.wiki/JPPN * Artikel Sprengel Museum https://www.wikidata.org/wiki/Q510144 Anschließend kann der Chatbot angewiesen werden, auf Grundlage der bereitgestellten Informationen bestimmte Aktionen auszuführen. Sie sollten den Chatbot bitten, Wikidata-SPARQL-Abfragen zu generieren, und diese Abfragen dann in die SPARQL-Abfrageoberfläche einfügen. https://query.wikidata.org/ Verwenden Sie diese Beispiele und entwickeln Sie Ihre eigenen: # Dashboard erstellen (Anzahl der Dinge) # Inventar erstellen (Tabelle) # Graphdatenmodell erstellen Einige SPARQL-Abfragen · Karte der Geburtsorte von Künstlern – https://w.wiki/JPT3 · Liste der Ausstellungen – https://w.wiki/JPR3 · Als Darstellung der Ausstellungen – https://w.wiki/J8aS ==== Hausaufgabe: Sitzung 2 ==== Erstellen Sie ein Bottom-up-Datenmodell eines Kunstwerks in einer Ausstellung. Fügen Sie nur die minimal erforderlichen Informationen hinzu. Das Ergebnis sollte eine Tabelle sein, wie sie für Ausstellung, Künstler und Katalog dargestellt wird. Die Tabelle sollte Eigenschaften und Attribute enthalten. Sie sollten die oben genannten Schemata zu Rate ziehen. Sie können KI verwenden, aber geben Sie die KI an und verlinken Sie sie mit Ihrer Frage. Wenn Sie KI verwenden, überprüfen Sie die Ergebnisse und machen Sie sich Notizen darüber, was Sie geändert haben. Hinweis: Überlegen Sie, wie die Teile miteinander in Beziehung stehen, was Sie hinzufügen müssen und was bereits in Wikidata vorhanden ist. Reichen Sie Ihre Ergebnisse als Tabelle oder Spreadsheet ein. --- ==== Sitzung 3: Museumsbesuch – Sprengel Museum ==== 19 März 2026 ENDE ==== Sitzung Nr. 4: Schemata und Prototyping (Abschlussprojekt) ==== ===== Zusammenfassung und Überblick ===== Erledigt * Erstellen von Ausstellungs-Einträgen in Wikidata * Befüllen unserer Datenmodelle für „Künstler“ und „Katalog“ * Erkundung des Museums und seiner Aktivitäten, um den Prototyp zu steuern Zu erledigen * Entscheidung über die Ideen für den Prototyp * Datenmodell für Objekte in einer Ausstellung (Kunstwerk und Ausstellung) * Erstellen eines Datenmodells zum Projektende, das von Museen genutzt werden kann und den Branchenstandards – CIDOC und Wikidata – entspricht. ===== Was haben wir über die „Geschichte des Museums“ gelernt? ===== TBC ===== Schemas ===== Eine Gelegenheit, sich mit der Struktur von Linked Open Data anhand gemeinsamer Vereinbarungen zu Arbeitspraktiken vertraut zu machen. Im Laufe des Kurses wird ein Datenmodell entwickelt und fertiggestellt, um „Objekte in einer Ausstellung“ zu beschreiben. Das Datenmodell wird zur Konsultation und zum Testen durch die Community veröffentlicht. ===== Schemas und Schlüsselkonzepte ===== Tabelle: https://tib.cloud/s/ZKNAAo3B8ATXsAP * Schema * Terminologiedienst * Kontrolliertes Vokabular * Taxonomie * Ontologie * Wissensgraph Tabelle X: Link: https://tib.cloud/s/ZKNAAo3B8ATXsAP In Linked Open Data (LOD) verwendete Terminologie DE {| class="wikitable" ! Konzept ! Wikidata-Link (Konzept) ! Hauptschwerpunkt ! Analogie ! Beispielressource ! URL ! Anwendungsbeispiel ! URL |- | Schema | Q1397073 | Datenstruktur | Die Vorlage. Konzeptionelles Schema / Datenmodell | Schema.org | [https://schema.org/] | VisualArtwork | [https://schema.org/VisualArtwork] |- | | | | | | | Smithsonian American Art Museum (SAAM) „In der Sierra Nevada, Kalifornien“ | [https://www.wikidata.org/wiki/Q20475372] |- | Terminologiedienst | Q22692845 | Verbreitung | Eine Bibliothek mit Vokabularen, Schemata, Ontologien usw. | TIB-Terminologiedienst | [https://terminology.tib.eu/ts/] | NFDI4CULTURE | [https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE] |- | Kontrolliertes Vokabular | Q1469824 | Konsistenz | Das Wörterbuch | Integrierte Normdatei / die Gemeinsame Normdatei (GND) | [https://portal.dnb.de/opac/showShortList] | Personen: Dürer, Albrecht | [https://d-nb.info/gnd/117751669] |- | Taxonomie | Q8269924 | Hierarchie | Sortierung nach Typ (allgemeine Klassifizierung) | Getty Art & Architecture Thesaurus (AAT) | [https://www.getty.edu/research/tools/vocabularies/aat/] | Deutscher Surrealist Max Ernst (verwendete Maltechniken) |[https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20und%20Grattage,in%20seinen%20Zeichnungen%20von%201925]. |- | | | | | Iconclass | [https://iconclass.org/] | Max Ernsts „Die Jungfrau, die das Christkind versohlt“ (Parady) | [https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926] |- | Ontologie | Q324254 | Semantik: Bedeutung & Logik (Informationswissenschaft) | Das Regelwerk oder der Stilführer | CIDOC (Comité International pour la DOCumentation / Internationales Komitee für Dokumentation) | [https://cidoc-crm.org/] | Sloane Lab Knowledge Base – Zusammenführung von 3 Sammlungen | [https://knowledgebase.sloanelab.org/resource/Start] |- | Wissensgraph | Q33002955 | Netzwerk von Dingen und Beziehungen | Eine Navigationskarte | Verzeichnis antiker Kunstwerke und architektonischer Bauwerke, die in der Renaissance bekannt waren | [https://www.census.de/] | Artemis-Suche | [https://database.census.de/#/detail/10013099] |- | | | | | Forschungsbereich | [https://researchspace.org/] | Hokusai: Das große Bilderbuch von allem |[https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start] |} EN {| class="wikitable" |- ! **Concept** ! **Wikidata link (Concept)** ! **Primary Focus** ! **Analogy** ! **Example resource** ! **URL** ! **Example use** ! **URL** |- | Schema | Q1397073 | Data Structure | The Template. Conceptual schema / data model | Schema.org | https://schema.org/ | VisualArtwork | https://schema.org/VisualArtwork |- | | | | | | | Smithsonian American Art Museum (SAAM) "Among the Sierra Nevada, California" | https://www.wikidata.org/wiki/Q20475372 |- | Terminology Service | Q22692845 | Distribution | A Library of Vocabularies, Schemas, Ontologies, etc | TIB Terminology Service | https://terminology.tib.eu/ts/ | NFDI4CULTURE | https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE |- | Controlled Vocabulary | Q1469824 | Consistency | The Dictionary | Integrated Authority File / die Gemeinsame Normdatei (GND) | https://portal.dnb.de/opac/showShortList | Persons: Dürer, Albrecht | https://d-nb.info/gnd/117751669 |- | Taxonomy | Q8269924 | Hierarchy | Sorting things by type (general classification) | Getty Art & Architecture Thesaurus (AAT) | https://www.getty.edu/research/tools/vocabularies/aat/ | German Surrealist Max Ernst (painting techniques used) | https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20and%20Grattage,in%20his%20drawings%20in%201925. |- | | | | | Iconclass | https://iconclass.org/ | Max Ernst’s "The Virgin Spanking the Christ Child" (Parady) | https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926 |- | Ontology | Q324254 | Semantics: Meaning & logic (information science) | The Rulebook or Writing Style Guide | CIDOC (Comité International pour la DOCumentation / International Committee for Documentation) | https://cidoc-crm.org/ | Sloane Lab Knowledge Base - unifying 3 collections | https://knowledgebase.sloanelab.org/resource/Start |- | Knowledge Graph | Q33002955 | Network of things and relations | A Navigational Map | Census of Antique Works of Art and Architecture Known in the Renaissance | https://www.census.de/ | Artemis search | https://database.census.de/#/detail/10013099 |- | | | | | Research Space | https://researchspace.org/ | Hokusai: The Great Picture Book of Everything | https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start |} ===== Schema-Übung ===== Zu bearbeitende Tabelle: https://tib.cloud/s/PicTdwCEqCQ6pBp (Passwort: bim2026) Wir werden uns mit folgenden Themen befassen: Ausstellung, Künstler und Katalog. '''''Geben Sie die gefundenen URLs ein. Fügen Sie bei Bedarf neue Zeilen, Spalten und Kommentare hinzu. Führen Sie sowohl manuelle als auch KI-Suchen durch, um die Ergebnisse zu vergleichen.''''' ===== Übung Nr. 1: Trage Links zu passenden Elementen aus den folgenden Quellen in die Tabelle ein: ===== * Wikidata:WikiProject Exhibitions/Properties * Generisches Wikibase-Modell für Kulturdaten – Wikibase4Research NFDI4Culture * CIDOC CRM (vollständig) * Terminologiedienst (NFDII4Culture) * Wikidata ===== Übung Nr. 2: Verwenden Sie KI-LLM, um passende Elemente zu finden ===== * <nowiki>https://gemini.google.com/</nowiki> ==== Prototyping ==== Entweder in dieser oder in der nächsten Sitzung wird die Gruppe in Teams aufgeteilt. ===== Schema ===== # Entwicklung eines Datenmodells: „Objekte in einer Ausstellung“ ===== Teile der Quarto-Publikation ===== # Ein Katalog einer Ausstellung des Sprengel Museums # Ein Katalog aller Ausstellungen und Ausstellungskataloge # Katalog der Ausstellungsbeiträge # Ein Glossar mit Begriffen – Personen und benannte Entitäten – aus Wikidata --- ===== Einführung in Quarto und Einfügen eines Ausstellungsbeitrags ===== Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace, Copilot: Agentic Coding) '''Voraussetzungen''' # Ein Laptop oder Computer, auf dem Sie VScode installieren können # Sie benötigen 2FA auf Ihrem Mobilgerät # Erstellen Sie ein GitHub-Konto # Installiere VScode # Verbinden Sie Ihr GitHub-Konto mit VScode # Erstellen Sie ein GitHub-Repository '''Klonen:''' https://github.com/mrchristian/prototype '''Modell: Auto''' '''So wurde das Repo eingerichtet. Agent-Eingabeaufforderungen:'''<blockquote>Ich möchte ein Quarto-Website-Projekt ausführen, bitte richte die Grundlagen ein. Das Projekt wird auf GitHub Pages veröffentlicht. Lege das Ausgabeverzeichnis auf „docs“ fest.</blockquote>Erstellen Sie eine Seite für das Quarto-Projekt, die die für diesen Wikidata-Eintrag verwendeten Daten abruft und als professionelle Webseite rendert <Fügen Sie hier Ihre Ausstellung ein – oder verwenden Sie diese> https://www.wikidata.org/wiki/Q138547468 Der Ansatz sollte eine SPARQL-Abfrage für die Daten erstellen und diese dann mithilfe eines Jupyter-Notebooks als HTML rendern. Alle Einträge: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Aufgaben ===== * Ausstellung ändern * Notebook ausführen * Quarto ausführen und Vorschau anzeigen * Auf Ihren GitHub Pages veröffentlichen ===== Schritt für Schritt ===== '''Teil 1: Arbeitsumgebung''' '''''HINWEIS: Sollten bei der lokalen Ausführung Probleme auftreten, nutzen Sie bitte die Online-Option von Codespace.''''' # Erstelle ein GitHub-Konto – https://github.com/ # Richte die Zwei-Faktor-Authentifizierung (2FA) ein – in der Regel auf dem Handy (Google Authenticator) # Installiere VSCode – https://code.visualstudio.com/download # Installiere GitHub Desktop – https://desktop.github.com/download/ # Füge dein GitHub-Konto hinzu, wenn du dazu aufgefordert wirst, und verwende die 2FA Schritt 2: Der Prototyp # Forken Sie das Repository: https://github.com/mrchristian/prototype # Wenn Sie lokal arbeiten, fahren Sie fort – wenn Sie Codespace verwenden, starten Sie Codespace (siehe unten und fahren Sie dann fort) # Testen Sie Quarto im Terminal: ## quarto check ## quarto render ## quarto preview (Strg+C – zum Beenden) # Falls es nicht funktioniert, führen Sie Quarto über den Agent aus # Ändern Sie die Wikidata-Ausstellung im Notebook # Notebook ausführen # quarto render und quarto preview ausführen # Alles speichern # Git: Nachricht, Commit und Push # Auf GitHub.com dein Repository ## Seiten aktivieren: GitHub Actions ## Code: Über das Zahnrad – Klicke auf „Meine GitHub Pages verwenden“ ## Registerkarte „Actions“: Quarto-Projekt veröffentlichen # ENDE – Wiederholen :-) ===== Codespace-Option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace ist eine virtuelle Maschine, die über GitHub gestartet werden kann. Das Repository enthält eine Dev-Container-Konfiguration, sodass du vollständig im Browser arbeiten kannst, ohne etwas lokal installieren zu müssen. 1.    Klicke auf der Repository-Seite auf GitHub auf „Code“ → „Codespaces“ → „Codespace erstellen“ auf der Hauptseite. 2.    Warte, bis der Container erstellt ist – Python-Pakete aus der Datei „requirements.txt“ werden automatisch installiert – dies dauert etwa 5 Minuten. 3.    Sobald alles installiert ist, kann der Codespace jederzeit genutzt werden. Er fährt automatisch herunter, wenn er nicht genutzt wird, und kann jederzeit neu gestartet werden. 4.    In Codespace geleistete Arbeit muss zurück ins Repository gepusht werden. 5.    Wenn Codespace 28 Tage lang nicht genutzt wird, wird der Codespace gelöscht. --- ===== Hausaufgabe – Sitzung Nr. 4 ===== * Hol alle Bücher aus der HsH-Bibliothek, die Ausstellungskataloge des Sprengel Museums sind. Bring sie zur nächsten Vorlesung mit * Erstelle einen Ausstellungs-Eintrag, falls noch nicht geschehen * Arbeite mit VSCode und dem Agent und experimentiere ==== Sitzung 5: Prototypenerstellung: Running Quarto Prototype, Federation, Prototype Teams ==== * Running Quarto Prototype - wie oben - https://github.com/mrchristian/prototype * DNB data download https://github.com/NFDI4Culture/linked-open-exhibition * Data Federation - WB4R ===== Links ===== https://wikibase.wbworkshop.tibwiki.io/wiki/Main_Page Glossar - https://nfdi4culture.github.io/linked-open-exhibition/Documentation/glossary ===== DNB Suche ===== https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books Sprengel Museum, 602 Artikel '''Über die Dienstleistungen von DNB''' https://www.dnb.de/librarylab https://deutsche-nationalbibliothek.github.io/jupyterlite/lab/ '''AUCH''' https://wiki.dnb.de/spaces/LINKEDDATASERVICE/pages/449878933/DNB+SPARQL+Service+BETA ===== Prototype Teams ===== * DNB-Daten * Katalog durchsuchen * Ausstellungsbeiträge * Vollständiges Datenmodell (alle) == Sitzung 6: Kursprojekt – Prototypenentwicklung: ''Vernetzte offene Ausstellungen'' == URL des Prototyps (derzeit 29.04.2026, ein Shell-Framework): https://nfdi4culture.github.io/linked-open-exhibition/ === Programm: === 11:30 – 11:50 Uhr (20 Min.) ''' Überblick: Klassenprojekt – Prototyping: Linked Open Exhibitions.''' Was ist das und was muss geliefert werden? Zuweisung zu Teilprojekten und Aufgabenübersicht. '''Aktivität Nr. 1: Bottom-up-Datenmodellierung: Datenabgleich''' 11:50 – 12:20 Uhr (30 Min.) Datenermittlung und -auswertung (Breakout-Räume) 12:20 – 12:40 Uhr (20 Min.) Besprechung der Datenergebnisse (Klassendiskussion) 12:40 – 12:55 (15 Min.) Pause '''Aktivität Nr. 2: Top-down-Datenmodellierung: Schema-Mapping''' 12:55 – 13:35 (40 Min.) Daten anhand von Schemata abbilden (Arbeitsgruppen) 13:35 – 13:55 (20 Min.) Besprechung der Ergebnisse (Klassendiskussion) '''13:55 – 14:15 (20 Min.) Arbeitszeit: Offener Zeitblock zur Überprüfung des laufenden „Tech Stack“ oder zur Beantwortung weiterer Fragen''' --- === Wichtige Links === ·  '''Haupt-Prototyp-Repository:''' https://nfdi4culture.github.io/linked-open-exhibition/ · Quarto-Einrichtung: ''„BIM Prototype 02 Quarto Website“:'' https://mrchristian.github.io/prototype/ ·  Anleitungen für den „Tech Stack“: ''Einführung in Quarto und Einfügen eines Ausstellungsbeitrags'' [[BIM-126-02-Data-Science-Linked-Open-Exhibition#Einführung in Quarto und Einfügen eines Ausstellungsbeitrags|https://en.wikiversity.org/wiki/BIM-126-02-Data-Science-Linked-Open-Exhibition#Einf%C3%BChrung_in_Quarto_und_Einf%C3%BCgen_eines_Ausstellungsbeitrags]] ·  Früherer Prototyp (2025): https://nfdi4culture.github.io/open-museum/ === ''Übersicht: Klassenprojekt – Prototypenentwicklung: Vernetzte offene Ausstellungen'' === Prototyp: https://nfdi4culture.github.io/linked-open-exhibition/ Repo: https://nfdi4culture.github.io/linked-open-exhibition/ Warum? ·      Rapid Prototyping wird in diesem Zusammenhang genutzt, um mehr über „Datenmodellierung mit Linked Open Data“ zu lernen. '''''Anmerkung: Die hier erworbenen Fähigkeiten und Erfahrungen im Bereich der Datenmodellierung sind eine Kernkompetenz, die eine Grundlage für die Erstellung von Datenmodellen in einer Vielzahl von beruflichen Kontexten bildet.''''' ◦  Wie man Datenmodellierung durchführt ◦  Zu verwendende Methoden: Bottom-up; KISS (Keep it Short and Simple); Top-down ◦  Bewertung und Validierung ◦  Ein Datenmodell operationalisieren ◦  Benutzertests ◦  Bewährte Verfahren, einschließlich Open-Science-Praktiken, z. B. die FAIR-Datenprinzipien ◦  Experimente mit KI-LLMs und agentischer Programmierung im Arbeitsablauf ·      Rapid Prototyping ist eine Design-Forschungsmethodik – das heißt, Wissen durch praktisches Tun zu schaffen oder zu entdecken. Was? Erstellen Sie mit der ganzen Klasse einen Website-Prototyp: https://nfdi4culture.github.io/linked-open-exhibition/ Die Website besteht aus drei datengesteuerten Teilprojekten: 1.    Manuelle Wikidata-Einträge für die Website des Sprengel Museums – Einträge der Klasse bereits erstellt 2.    Massen-Ausstellungseinträge, abgeleitet aus über 600 DNB-Einträgen zum „Sprengel Museum“ – importiert 3.    HsH-Bibliotheksdatensätze für eine Suche zum Sprengel Museum und ein Scan eines Sprengel-Museum-Katalogs für Text- und Daten-Mining (TDM) – noch zu erledigen Wie? Simon Worthington wird als Publikationsmanager fungieren. Dies umfasst die Ausführung oder Steuerung komplexer Softwareteile. Für einige Teile wird agentische Codierung mit Copilot verwendet (erprobt). Die Klasse ist in drei Teams für die Teilprojekte aufgeteilt: 1.    Website zu den Ausstellungen des Sprengel Museums; 2.    DNB-Einträge „Sprengel Museum“ 3.    Text- und Data-Mining: Bibliothekskatalog Sprengel Museum Jedes Team führt für seinen Teil die gleichen Aufgaben durch, um eine Runde der Datenmodellierung abzuschließen: 1.    Daten sammeln – Bottom-up-Methode 2.    Validierung der Daten – Top-down-Methode 3.    Präsentation der Daten – Quarto-Website „Linked Open Exhibitions“ Ziel: Definition of Done (DoD) ''Anmerkung: Entwicklerjargon'' ·      Ein dokumentiertes Datenmodell (Tabelle) mit Diagramm (Mermaid, GraphVis oder Draw.io) ·      Zuordnung des Datenmodells zu Schemata (Tabelle) ·      Idee zur Darstellung der Daten für jeden Unterabschnitt im Prototyp und Umsetzung mit Unterstützung des Publication Managers, z. B. für DNB eine chronologische Liste von Ausstellungen mit Bildern. ·      Dokumentation der Nutzung von KI-LLM als Assistent, Quellenangaben und Kommentare zu bewährten Verfahren ·      Datenherkunft und Ausfüllen der Checkliste für bewährte Praktiken ·      Das Endergebnis „Class Project – Prototyping: Linked Open Exhibitions“ wird als institutionelle Hinterlegung bei [https://zenodo.org/ Zenodo] veröffentlicht. === --- === === Aktivität Nr. 1: Datenerhebung und Bottom-up-Datenmodellierung === Bestätigen, erstellen oder erweitern Sie bestehende Datenmodelle, indem Sie die Quelle betrachten. Jedes Projekt hat eine Quelle: * ·      Team Nr. 1: Website des Sprengel Museums, Ausstellungslisten: ◦  https://www.sprengel-museum.de/ und ◦  CSV-Tabelle mit den eingegebenen Ausstellungen GitHub | [https://tib.cloud/s/fncf8W6pXs8qgiq Tabelle TIB Cloud] (passwortgeschützt) ◦  Prototyp: https://nfdi4culture.github.io/linked-open-exhibition/exhibitions.html * ·      Team Nr. 2: DNB-Einträge zur Suche nach „Sprengel Museum“: ◦  https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books | https://wikibase.wbworkshop.tibwiki.io/ ◦  CSV https://github.com/NFDI4Culture/linked-open-exhibition/blob/main/catalogues/sprengel_exhibitions.csv ◦  Bilder von Buchumschlägen https://github.com/NFDI4Culture/linked-open-exhibition/tree/main/catalogues/images * ·       Team Nr. 3: Informationen der HsH-Bibliothek zu Katalogen für das „Sprengel Museum“ : ◦  https://katalog.bib.hs-hannover.de/vufind/Search/Results?lookfor=Sprengel%2BMuseum ◦  Team 3 muss bei Null anfangen, da wir bislang noch keinen Bibliotheksdatensatz oder Eintrag in einem Ausstellungsdatenmodell haben. Tipp: Schaut euch die anderen Modelle an, um mit dem Aufbau eurer Datenmodelle zu beginnen. '''>>> Hier zum Datenmodell hinzufügen:''' https://tib.cloud/s/PicTdwCEqCQ6pBp (passwortgeschützt) ==== ZIEL ==== Sicherstellen, dass das Datenmodell die Quelle abbilden kann. Gibt es genügend Einträge, um die Bestandteile der Quelle zu beschreiben? Der Prozess ist iterativ, das heißt, er wird immer wieder wiederholt, wobei Verbesserungen und Änderungen vorgenommen werden. ==== AUFGABEN ==== ·      Bearbeiten Sie den violett-grauen Bereich; der grüne Bereich wird in der nächsten Aktivität bearbeitet ·      Überprüfen und korrigieren Sie vorhandene Informationen ·      Fügen Sie neue Konzepte hinzu, falls die Quelle dies erfordert ·      Orangefarbene Bereiche müssen ausgefüllt werden. Die Zellen müssen möglicherweise bearbeitet oder ergänzt werden. ·      Datentypen finden Sie ausschließlich auf den Eigenschaftsseiten. Elemente (QIDs) haben keine Datentypen, unter „https://www.wikidata.org/wiki/Property:P1476<nowiki/>“ unter der Bezeichnung „ “ ''Datentyp'' ·      URI entspricht einer URL === Tipps === * Schau dir andere Beispiele auf Wikidata an: Künstler, Ausstellungen, Kataloge, bibliografische Einträge oder Objekte in einer Ausstellung. * Verwenden Sie eine KI, um Schemaerklärungen oder Optionen nachzuschlagen. Registrieren Sie sich bei KISSKI, um eine bessere KI-Datenschutznutzung zu erhalten. === Aktivität Nr. 2: Validierung der Top-Down-Datenmodellierung === ·      Team 1: Website des Sprengel Museums ·      Team 2: DNB-Einträge zur Suche nach „Sprengel Museum“   ·      Team 3: Informationen der HsH-Bibliothek zu Katalogen zum „Sprengel Museum“ ==== ZIEL ==== Alle Konzepte abbilden ==== AUFGABEN ==== Das Konzept in den verschiedenen Ressourcen nachschlagen und Verknüpfungen hinzufügen, === Arbeitszeit: Zeitfenster zur Überprüfung des laufenden „Tech Stack“ oder zur Klärung sonstiger Fragen === --- == Hausaufgabe == ·      Führen Sie die Bottom-up- und Top-down-Modellierung durch ·      Team #3: Besuche die Bibliothek und erstelle einen digitalen Scan auf einem Kopierer, speichere ihn als PDF. Der Scan wird für Text- und Data-Mining verwendet und die Datei anschließend gelöscht und vernichtet. Wir werden nur Metadaten aus dem Scan extrahieren. ·      Kommt zur nächsten Unterrichtsstunde mit Ideen und Vorschlägen, was ihr aus euren Daten und Datenmodellen im Prototyp dargestellt haben möchtet. ENDE --- == EN == ''Materials and Tasks for the module "BIM-126-02, SoSe 2026, Worthington/Blümel" for students at Hochschule Hannover. The materials are prepared with several colleagues from the [https://www.tib.eu/de/forschung-entwicklung/forschungsgruppen-und-labs/open-science Open Science Lab at TIB] Hannover.'' Project GitHub repo: https://github.com/NFDI4Culture/linked-open-exhibition ==== Summary ==== The eight session course covers an introduction to Linked Open Data (LOD) in the context of : # Open Galleries Libraries Archives and Museums (GLAM), and; # The use of Wikimedia Foundation platforms. The Wikimedia Foundation platforms that will be used are: Wikidata; Wikibase, MediaWiki, and Wikimedia Commons. AI LLM will be used in the workflows: Code assistant ''copilot'', and a variety of AI LLM chat services for file generation and configurations to create SPARQL queries, Jinja 2.0 templates, etc. „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat The Methodologies employed are: Open-source software, Open Science, and rapid prototyping. ==== Linked Open Exhibition ==== The question being explored for the class is how can LOD be uséd to benefit museum exhibitions as Linked Open Exhibitions – a record of the exhibition, a catalogues of items in an exhibition, and other important data? As examples '''to gain exhibitions increased visitors numbers and create greater depth of engagement'''. With a focus of the question on how to make LOD records of '''items in an exhibition'''. ==== Learning points – In order of priority ==== # '''Wikidata/Wikibase LOD concepts:''' Items, Properties, Values, Qualifiers, Wikibase schemas, Classes, Lexemes, Knowledge Base, and Knowledge Graphs. # '''Linked Open Data (LOD):''' Semantic web, 5 star, RDF/Triples, Ontologies, Taxonomies, and controlled vocabularies. # '''Using LOD source:''' Identifiers, PIDs, information sources, media sources, and import and export tooling. # '''Data modelling:''' Methodologies, schema use, visualisation, and testing. # '''Data workflow tools:''' Git, IDE, AI code assistant (copilot), AI Chat, using Wikimedia Foundation tooling, data import and export tools, generating PIDs and making deposits in a scholarly repository. # '''Data presentation and data use:''' Wikidata Query Service results, MediaWiki infoboxes, AI Chat SPARQL query processing. # '''Open Science practice:''' Open-source software, Open Notebook Science, Open Licencing, PIDs, FAIR Data Principles, and ethical and good practice AI use. ==== Sessions ==== The sessions would be about cataloguing Sprengel Museum exhibitions using LOD and how to make visualisations and presentations. '''Learning to use LOD is the goal of the learning.''' The method will be to build out from a kernel of an ‘exhibition’ and add ‘item in an exhibition’. From the start the students will be the ones who make the LOD. This will start with minimal entries my by the students, then layering these up with – Identifiers, LOD Media sources, schemas, etc. And finally moving onto how to present the data in a way that satisfies the ‘use case’: '''to gain exhibitions increased visitors numbers and create greater depth of engagement'''. Here presentation technologies are used: MediaWiki infoboxes, Wikidata Query Service results, AI Chat SPARQL queries and other features, etc. ===== Session 1: Exhibition timeline creation - build out, add exhibitions ===== # Record minimal information for an exhibition in Wikidata as Linked Open Data: Title, museum, date, etc. e.g., https://www.wikidata.org/wiki/Q138547468 – See: Table 1: ''Minimal data entries for an exhibition'' # View the exhibition record in Wikidata Query Service results link (timeline and graph https://w.wiki/J8NJ | https://w.wiki/J8aS ) # Review exhibition entries. # Cover topics raised by making a LOD entry: Wikidata basics, Wikidata good practice, consulting schemas, importance of review and using GitHub Issues, comparing available data – before and after. ===== Session 2: Exhibition cataloguing - build up, add items, artists, catalogues ===== ===== Session 3: Museum visit - Sprengel Museum ===== ===== Session 4: Schemas and Prototyping (the end of class project) ===== ===== Session 5: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 6: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 7: Prototype Creation: Data entry, visualisation, and presentation ===== ===== Session 8: Prototype Creation: Data entry, visualisation, and presentation ===== --- ==== Session 1: Exhibition timeline creation - build out, add exhibitions ==== The exercise: Create a Linked Open Data record for an exhibition using Wikidata (minimal entry). A. '''Creating the exhibition entry in Wikidata.''' # Login to Wikidata: https://www.wikidata.org/ # Have a source at hand to make a data entry, e.g., #* https://www.sprengel-museum.de/ausstellungen/archiv #* https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised #* https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&currentPosition=1 # Check there is no existing entry for the exhibition is on Wikidata. Use the search function. # Create an item or edit an existing item. #* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch). # Create the following data entries in Wikidata, see: Table 1: ''Minimal data entries for an exhibition.'' # Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., #* All entries present [ ] #* All entries correct [ ] #* Entries are in Deutsch and English – within reason [ ] ''Table'' ''1: Minimal data entries for an exhibition'' {| class="wikitable" | colspan="7" |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468''' |- |A |Label | colspan="5" |Note: Keep short. Use title from exhibition |- |B |Description | colspan="5" |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026 |- | |'''Property (P) and Item (Q)''' |'''URI''' |'''DE''' |'''EN''' |'''Add''' |'''Note''' |- |1 |P31 |https://www.wikidata.org/wiki/Property:P31 |ist ein(e) |instance of |Q464980 |Add item |- |2 |Q464980 |https://www.wikidata.org/wiki/Q464980 |Ausstellung |Exhibition | |(Used above) |- |3 |P1476 |https://www.wikidata.org/wiki/Property:P1476 |Titel |Title |Title |Plain text |- |4 |P276 |https://www.wikidata.org/wiki/Property:P276 |Ort |Location |Sprengel Museum Hannover Q510144 |Add item |- |5 |P580 |https://www.wikidata.org/wiki/Property:P580 |Startzeitpunkt |Start time |Date |YYYY-MM-DD |- |6 |P582 |https://www.wikidata.org/wiki/Property:P582 |Endzeitpunkt |End time |Date |YYYY-MM-DD |- |7 |P1640 |https://www.wikidata.org/wiki/Property:P1640 |Kurator |Curator |Person |Add item (if don't exists will need to create/can omit at present) |- |8 |P710 |https://www.wikidata.org/wiki/Property:P710 |Teilnehmer |Participant |Person (the artist) |Add item (if don't exists will need to create/can omit at present) |- |9 |P856 |https://www.wikidata.org/wiki/Property:P856 |offizielle Website |Official website |URL |URL |} '''''End of Session 1.''''' ==== Homework exercises ==== # Complete your allocated exhibition. Make sure all fields are complete from Table 1. If something cannot be added, either: A. Make a note in the exhibition allocation spreadsheet, or B. Send and email to [mailto:Simon.worththington@tib.eu simon.worththington@tib.eu] and I will help resolve your issue. '''Note: If you did not create an exhibition entry during the class make sure one is complete before the next class.''' # Create a GitHub account and add your GitHub handle next to your name, column ‘GitHub handle’, in the exhibition allocation spreadsheet. # Review your classmates exhibition entries. You have all been allocated a entry to review, see the Exhibition Allocation spreadsheet. Your name will be in column G. This first review has three questions – tick the boxes to show if each item has been complete and either add comments or correct the Wikidata exhibition entry. '''Note: If your allocated Exhibition entry hasn’t been made by you classmate then please contact them and ask them to complete the entry.''' Questions are: ## Are all the required fields present? ## Are all the fields correct? ## Is there an Deutsch and English entry? --- ==== Session 2: Exhibition cataloguing - build up, add items, artists, catalogues ==== The session has five exercies: # Exhibition update # Artist # Exhibition catalogue # AI LLM SPARQL experiments # <s>Artwork</s> The exercises include the following concepts: ==== Exercises ==== ==== 1. Exhibition updates ==== * Homework review: Complete all fields for an exhibition. Review your assigned review exhibition answering the three questions: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch</blockquote> * For the label. Convert words in all caps to sentence case. Use: https://convertcase.net/title-case-converter/ | Change from, e.g., ADRIAN SAUER: TRUTH TABLESPECTRUM INTERNATIONALER PREIS FÜR FOTOGRAFIE DER STIFTUNG NIEDERSACHSEN to Adrian Sauer: Truth Tablespectrum Internationaler Preis Für Fotografie Der Stiftung Niedersachsen. * Add the English language versions. Use DeepL to translate: https://www.deepl.com/en/translator ** Title: Add English title * Add the following. Change P710 Teilnehmer (Participant) to P921 zentrales Thema '''artists name.''' ** Qualifier on central theme to indicate the person is contributing artwork. * Use: Qualifier P170 creator and add artist Q483501 (type artists and it will automcomplete) * Reference: Gemeinsame Normdatei (GND) ID for a person, e.g., Gabriela Jolowicz https://d-nb.info/gnd/134184963 | Search your persons name and copy in the last part of number 134184963 * Talk page: Add in the review questions for your Wikidata entry: <blockquote>[ ] Sind alle erforderlichen Felder vorhanden? [ ] Sind alle Felder korrekt ausgefüllt? [ ] Gibt es einen Eintrag in Deutsch und Englisch?</blockquote>Notice the useful links that tell you more about connected Linked Open Data! Note: SPARQL query showing data model. Properties and and values. Results: https://w.wiki/JMLX Made with Gemini AI: https://gemini.google.com/share/c43f34a67f67 ==== Concepts ==== * Wikidata parts – see about and diagram: ** https://www.wikidata.org/wiki/Wikidata:Introduction/de ** https://www.wikidata.org/wiki/Wikidata:Introduction#/media/File:Datamodel_in_Wikidata.svg * Applying a review process using Talk pages * Adding References * Using a type of LOD source – '''An authority record''' Gemeinsame Normdatei (GND) ID https://portal.dnb.de/opac.htm * SPARQL query --- ==== 2. Artists ==== The objective here is to ensure all artists have been included in exhibition listing and to then review the existing artists entry. Later a SPARQL query will be made to compare statements about all the artists in our dataset. * Before reviewing artists items make sure all artists have been listed in the exhibition item, with qualifier of being an artist and a reference to their GND record. ===== Important statements ===== {| class="wikitable" |Concept |CIDOC CRM (Full) |Linked Art (Selection) |Wikidata Equivalent |Note |- |Entity |E21 Person |Person |Q5 (human) |The base instance. |- |Label/Name |P1 is identified by → E33_E41 |identified_by (Name) | |Linked Art flattens this into a simple list of names. |- | | | |P735 Given name | |- | | | |P734 Family name | |- |Profession |P2 has type → E55 Type |classified_as |P106 (occupation) |Map to AAT 300025103 (artist). |- |Birth |P98i was born → E67 Birth |born (Birth) |P569 (date of birth) |CRM treats birth as an event; Wikidata as a property. |- |Death |P100i died in → E69 Death |died (Death) |P570 (date of death) |If the artist is still living, this is omitted. |- |Nationality |P107i member of → E74 Group |classified_as (Type) |P27 (citizenship) |Linked Art often models nationality as a Type. |- |Reference |P1 identifies ← E42 Identifier |identified_by (Identifier) |QID (The URI itself) |Used to link to external authorities (ULAN, VIAF). |- |Commons category |? |? |P373 search name |<nowiki>https://commons.wikimedia.org/</nowiki> |} From Google Gemini: https://gemini.google.com/share/578cc1b886d0 --- ===== Schemas and communities need consulting. ===== From Wikimedia: * WikiProject Visual Arts: https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Visual_arts * Wikiproject Exhibitions: https://www.wikidata.org/wiki/Wikidata:WikiProject_Exhibitions Semi-formal Generic Wikibase Model for Cultural Data: https://kgi4nfdi.github.io/Guidelines/guide/wikibase/data_modelling_import/ Formal: CIDOC Conceptual Reference Model (CRM) - https://cidoc-crm.org/ Linked Art (based on CIDOC) https://linked.art/model/actor/ ==== Concepts ==== * Data modeling * Schemas * Use case * Bottom up design * Identifiers --- ==== 3. Exhibition Catalogue ==== Search in both of these two places to find information about the catalogue for your assigned exhibition. * Sprengel Museum publication catalogue - https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised * DND (example) you can search for the exhibition name or Sprengel Museum '''-''' https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&cqlMode=true ''Note: Make a note of any links you find in the exhibition listings spreadsheet.'' ===== Make a Wikidata entry for the catalogue ===== Note: first search for publication before making Wikidata entry. Use title, use ISBN, use GND. An example publication from DNB and Sprengel Shop. * https://portal.dnb.de/opac/showFullRecord?currentResultId=Gabriela+and+Jolowicz%26any&currentPosition=0 * https://www.sprengel-museum.de/besuch?view=article&id=65:publikationen&catid=2:uncategorised ===== Enter these statements ===== Note: Remember Label and Description {| class="wikitable" |Property |Label |Description/Example |- |P31 |instance of |catalogue (Q2352616) |- |P1476 |title |The official title of the catalogue (e.g., Vermeer and the Masters of Genre Painting) |- |P50 |author |The main curator or art historian (item link) |- |P123 |publisher |The museum or publishing house (e.g., Louvre Museum) |- |P577 |publication date |Year of release (e.g., 2024) |- |P212 |ISBN-13 |The 13-digit standard book identifier |- | |GND |ID |- |P973 |described at URL |A link to the catalogue's page on the museum’s website |} Google Gemini https://gemini.google.com/share/9a21f5522192 Example input: https://www.wikidata.org/wiki/Q138646145 ===== Link the record back to the exhibition ===== P972 Title ==== Concepts ==== * Data modeling * Identifier * Data as CC Zero / Copyright of data --- ==== 4. AI LLM SPARQL experiments ==== The Wikidata has a SPARQL interface where the LOD in Wikidata can be searched (queried) and outputted in a number of ways, formats, and a visualisations. As well as being saved on the web. We will us AI LLM chat to generate SPARQL queries. Later we will learn the fundamentals of writing a SPARQL query. But for the moment we want to see how they have be generated, the options, and creative applications. Using chat services or code assistants can be a valuable way to learn about new technologies. {| class="wikitable" |Service |Best For |Standout Feature |Key Model(s) |- |'''ChatGPT''' |General Use & Tasks |Deep Research & Agent Mode |GPT-5.4, GPT-5 |- |'''Claude''' |Coding & Writing |Artifacts (interactive workspace) |Claude 4.5, 4.6 |- |'''Google Gemini''' |Google Ecosystem |Nano Banana (native image/video) |Gemini 3.1 Pro |- |Perplexity |Real-time Research |Native Citations & Search Labs |Sonar, GPT-5, Claude |- |MS Copilot |Office Productivity |Copilot Vision & 365 Integration |GPT-5.2, Prometheus |- |DeepSeek |Logical Reasoning |High-tier performance at low cost |DeepSeek-V3, R1 |- |Grok |Real-time Social Info |Unfiltered X (Twitter) integration |Grok 4.1 |- |'''Meta AI''' |Social Media |Seamless integration in WhatsApp/IG |Llama 4 (Scout) |- |Poe |Model Testing |Access multiple LLMs in one app |Multi-model aggregator |- |Mistral (Le Chat) |Privacy & Developers |European-hosted, GDPR-focused |Mistral Large 3 |} Some of these can also be used via KISSKI „KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 https://kisski.gwdg.de/leistungen/2-02-llm-service/ | https://chat-ai.academiccloud.de/chat  === The exercise === The group will be split into a number of Zoom breakout groups and then the group spends 20 minutes experimenting generating SPARQL queries and other creative applications. Paste in results here: https://tib.cloud/apps/files/files/8251374?dir=/NFDI4Culture/HsH/BIM26/bim26-shared&editing=false&openfile=true Each room is assigned a Chat engine. Maximum there will be four groups. ·      Group #1: '''ChatGPT''' ·      Group #2: '''Claude''' ·      Group #3: '''Google Gemini''' ·       Group #4: '''Meta AI''' === Example exercise === Chat bots can read  a SPARQL query or a Wikidata address. e.g., Item https://www.wikidata.org/wiki/Q138547468 query graph https://w.wiki/JPNc query timeline https://w.wiki/JPPN Item Sprengel Museum https://www.wikidata.org/wiki/Q510144 Then the chatbot can be instructed to do things based on the information provided. You should ask the chat bot to generate Wikidata SPARQL queries and then paste the queries into the SPARQL querie interface. https://query.wikidata.org/ Use these examples and invent your own: # Create dashboard (count of things) # Create inventory (table) # Create graph data model Some output SPARQL queries ·       Map of artists place of birth - https://w.wiki/JPT3 ·       List of exhibitions - https://w.wiki/JPR3 ·       As plot of exhibitions - https://w.wiki/J8aS ==== Homework: Session 2 ==== Create a bottom up data model of an artwork in an exhibition. Include only the minimum information needed. The result should be a table like the ones presented for exhibition, artist, and catalogue. The table should include properties and attributes. You should consult the schemas mentioned above. You can use AI but attribute the AI and link to your question. If you use AI review the results and make notes about what you changed. Note: Think about how parts are related and what you need to add and what already exists in Wikidata. Submit your results as a spreadsheet or table. ===== Session 3: Museum visit - Sprengel Museum ===== 19 March 2026 ===== Session #4: Schemas and Prototyping (the end of class project) ===== ===== Recap and outline ===== Done * Creating exhibition entries in Wikidata * Filling our data models for Artist and Catalogue * Exploring the museum and its activities to help steer the prototype To do * Decide on the ideas for the prototype * Data model for items in an exhibition (Artwork and Exhibition) * Complete a data model for the end of the project that can be used by museums and complies to the sector standards – CIDOC and Wikidata. ===== What have we learned about the ‘Museum’s Story’ ===== TBC ===== Schemas ===== An opportunity to become familiar with how Linked Open Data is structured using common agreements on working practices. Over the period of the course a data model will be developed and finalised to describe ‘items in an exhibition’. The data model will be published for community consultation and testing. ===== Schemas and key concepts ===== Table: https://tib.cloud/s/ZKNAAo3B8ATXsAP * Schema * Terminology Service * Controlled Vocabulary * Taxonomy * Ontology * Knowledge Graph Table X: link: https://tib.cloud/s/ZKNAAo3B8ATXsAP Terminology used in Linked Open Data (LOD) {| class="wikitable" |- ! **Concept** ! **Wikidata link (Concept)** ! **Primary Focus** ! **Analogy** ! **Example resource** ! **URL** ! **Example use** ! **URL** |- | Schema | Q1397073 | Data Structure | The Template. Conceptual schema / data model | Schema.org | https://schema.org/ | VisualArtwork | https://schema.org/VisualArtwork |- | | | | | | | Smithsonian American Art Museum (SAAM) "Among the Sierra Nevada, California" | https://www.wikidata.org/wiki/Q20475372 |- | Terminology Service | Q22692845 | Distribution | A Library of Vocabularies, Schemas, Ontologies, etc | TIB Terminology Service | https://terminology.tib.eu/ts/ | NFDI4CULTURE | https://terminology.tib.eu/ts/ontologies?and=false&page=1&sortedBy=title&size=10&collection=NFDI4CULTURE |- | Controlled Vocabulary | Q1469824 | Consistency | The Dictionary | Integrated Authority File / die Gemeinsame Normdatei (GND) | https://portal.dnb.de/opac/showShortList | Persons: Dürer, Albrecht | https://d-nb.info/gnd/117751669 |- | Taxonomy | Q8269924 | Hierarchy | Sorting things by type (general classification) | Getty Art & Architecture Thesaurus (AAT) | https://www.getty.edu/research/tools/vocabularies/aat/ | German Surrealist Max Ernst (painting techniques used) | https://www.guggenheim-venice.it/en/art/conservation-department-new/technical-studies-and-conservation-campaigns/portrait-of-an-artist-at-work-max-ernsts-surrealist-techniques/#:~:text=Frottage%20and%20Grattage,in%20his%20drawings%20in%201925. |- | | | | | Iconclass | https://iconclass.org/ | Max Ernst’s "The Virgin Spanking the Christ Child" (Parady) | https://www.wikiart.org/en/max-ernst/the-virgin-spanking-the-christ-child-before-three-witnesses-andre-breton-paul-eluard-and-the-1926 |- | Ontology | Q324254 | Semantics: Meaning & logic (information science) | The Rulebook or Writing Style Guide | CIDOC (Comité International pour la DOCumentation / International Committee for Documentation) | https://cidoc-crm.org/ | Sloane Lab Knowledge Base - unifying 3 collections | https://knowledgebase.sloanelab.org/resource/Start |- | Knowledge Graph | Q33002955 | Network of things and relations | A Navigational Map | Census of Antique Works of Art and Architecture Known in the Renaissance | https://www.census.de/ | Artemis search | https://database.census.de/#/detail/10013099 |- | | | | | Research Space | https://researchspace.org/ | Hokusai: The Great Picture Book of Everything | https://hokusai-great-picture-book-everything.researchspace.org/resource/rsp:Start |} ===== Schemas exercise ===== Spreadsheet to work on: https://tib.cloud/s/PicTdwCEqCQ6pBp (password: bim2026) We will be looking at: Exhibition, Artist, and Catalogue. '''''Enter the URLs found. Add new rows, columns, comments if needed. Keep manual searches as well as AI searches for comparison.''''' ===== Exercise #1: Enter links into the spreadsheet of matching items from the following: ===== * Wikidata:WikiProject Exhibitions/Properties * Generic Wikibase Model for Cultural Data - Wikibase4Research NFDI4Culture * CIDOC CRM (Full) * Terminology Service (NFDII4Culture) * Wikidata ===== Exercise #2: Use AI LLM to find matching items ===== * https://gemini.google.com/ ==== Prototyping ==== Either in this session or in the next session the group will be divided into teams. ===== Schema ===== # Data model development: ‘items in an exhibition’ ===== Quarto publication parts ===== # A catalogue of a Sprengel Museum exhibition # A catalogue of all exhibitions and exhibition catalogues # Catalogue of exhibition entries --- ==== Learning to use Quarto and inserting an exhibition entry ==== Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding) '''Requirements''' # A laptop or computer where you can install VScode # You will need 2FA on your mobile (optional) # Create a GitHub account # Install VScode # Connect Github account to VScode # Create GitHub reposoitory '''Fork the following repository:''' https://github.com/mrchristian/prototype '''Model: Auto''' '''How the repo was setup. Agent promts:'''<blockquote>''I want to run a Quarto website project, please setup the basics. The project will be published on GitHub Pages. Set the output directory to docs.'' </blockquote>Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as professional webpage ''<Insert your exhibition here – or use this one>''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook. All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password) ===== Tasks ===== * Change exhibition - manual * Run Jupyter Notebook * Run and preview Quarto * Publish to your GitHub Pages ===== Step-by-step ===== ====== Part one: Working environment ====== '''''NOTE: If you are having problems running locally then use the Codespace online option.''''' # Create GitHub account - https://github.com/ # Have 2FA available - usually on mobile (Google authenticator) (optional) # Install VSCode - https://code.visualstudio.com/download # Install GitHub Desktop - https://desktop.github.com/download/ # Add Github account when prompted, use 2FA ====== Step two: The prototype ====== # Fork the repository: https://github.com/mrchristian/prototype # If working locally continue - if using Codespace - launch Codespace (see below and then continue) # Test Quarto in the Terminal: ## <code>quarto check</code> ## <code>quarto render</code> ## <code>quarto preview</code> (control C - to stop) # If not working run Quarto from Agent # Change Wikidata exhibition in Notebook # Run notebook # Run <code>quarto render</code> <code>quarto preview</code> # Save all (or use auto save) # Git: Message, Commit and Push # On GitHub.com your repository ## Turn on Pages: GitHub Actions ## Code: About cog - Click use my GitHub Pages ## Actions tab: Publish Quarto Project # ENDE - Rinse repeat :-) ===== Codespace option: ===== Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit) Codespace is an online Virtual Machine which can be launched from GitHub. The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally. # On the repository page on GitHub, click Code → Codespaces → Create codespace on main. # Wait for the container to build — Python packages from <code>requirements.txt</code> are installed automatically - about 5 minutes. # Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time. # Work done in Codespace must be pushed back to the repository. # If Codespace is not used for 28 days the Codespace is deleted. --- ===== Homework - session #4 ===== * Get all books from HsH library that are Sprengel Museum exhibition catalogues. Bring to the next class * Make an exhibition entry if not done * Work with VSCode and the Agent and experiment * Add entries from existing ontologies: https://tib.cloud/s/PicTdwCEqCQ6pBp?dir=/&editing=false&openfile=true ==== Sitzung 5: Prototyping: Running Quarto Prototype, Federation, Prototype Teams ==== * Running Quarto Prototype - https://github.com/mrchristian/prototype * DNB data download https://github.com/NFDI4Culture/linked-open-exhibition * Data Federation - WB4R ===== Links ===== https://wikibase.wbworkshop.tibwiki.io/wiki/Main_Page Glossar - https://nfdi4culture.github.io/linked-open-exhibition/Documentation/glossary ===== DNB Search ===== https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books Sprengel Museum, 602 Artikel '''About the DNB''' https://www.dnb.de/librarylab https://deutsche-nationalbibliothek.github.io/jupyterlite/lab/ '''Also''' https://wiki.dnb.de/spaces/LINKEDDATASERVICE/pages/449878933/DNB+SPARQL+Service+BETA ===== Prototype Teams ===== * DNB Data * Katalog scan * Exhibition entries * Datenmodell (alle) ===== --- ===== == Session 6: Class Project – Prototyping: ''Linked Open Exhibitions'' == Prototype URL (currently 2026-04-29 a shell framework): https://nfdi4culture.github.io/linked-open-exhibition/ ==== Program: ==== 11:30 – 11:50 (20 min) '''Outline: Class Project – Prototyping: Linked Open Exhibitions.''' What is it and what needs to be delivered. Allocation to sub-project and tasks outline. '''Activity #1: Bottom-up data modeling: Data mapping''' 11:50 – 12:20 (30 min) Data finding and exploration (break out rooms) 12:20 – 12:40 (20 min) Review data findings (class discussion) 12:40 – 12:55 (15 min) Pause Break '''Activity #2: Top-down data modeling: Schema mapping''' 12:55 – 13:35 (40 min) Map data against schemas (break out rooms) 13:35 – 13:55 (20 min) Review findings (class discussion) '''13:55 – 14:15 (20 min) Work time: Open time-slot to review running ‘Tech Stack’ or address any other questions''' --- ==== Important links ==== * '''Main Prototype repository:''' https://nfdi4culture.github.io/linked-open-exhibition/ * Quarto setup: ‘''BIM Prototype 02 Quarto Website’:'' https://mrchristian.github.io/prototype/ * Instructions for ‘Tech Stack’: ''Einführung in Quarto und Einfügen eines Ausstellungsbeitrags'' [[BIM-126-02-Data-Science-Linked-Open-Exhibition#Einführung in Quarto und Einfügen eines Ausstellungsbeitrags|https://en.wikiversity.org/wiki/BIM-126-02-Data-Science-Linked-Open-Exhibition#Einf%C3%BChrung_in_Quarto_und_Einf%C3%BCgen_eines_Ausstellungsbeitrags]] * Earlier Prototype (2025): https://nfdi4culture.github.io/open-museum/ ==== ''Outline: Class Project – Prototyping: Linked Open Exhibitions'' ==== Prototype: https://nfdi4culture.github.io/linked-open-exhibition/ Repo: https://nfdi4culture.github.io/linked-open-exhibition/ Why? * Rapid Prototyping in this context is used to learn about ‘Data Modeling using Linked Open Data’. '''''NB: The data modeling skills and experience learned here is a core competence that gives a foundation to be able to create data models in a wide set of professional contexts.''''' ** How to do data modeling ** To use method: Bottom-up; KISS (Keep it Short and Simple); Top-down ** Evaluation and validation ** Operationalize a data model ** User testing ** Good practice, including Open Scholarship (Open Science) practice. e.g. FAIR Data Principles ** Experiment with AI LLMs and agentic coding in the workflow * Rapid Prototyping is a Design Research methodology – meaning to create or discover knowledge by doing. What? Create a website prototype a the whole class: https://nfdi4culture.github.io/linked-open-exhibition/ The website is made of three data driven sub-projects: # Manual Wikidata entries for Sprengel Museum website – class entries already made # Bulk exhibition entries derived from 600+ DNB records for ‘Sprengel Museum’ - imported # HsH Library information records for a search on the Sprengel Museum and one scan of a Sprengel Museum catalogue for Text and Data Mining (TDM) – to do How? Simon Worthington will act as Publication Manager. This involves running or guiding complex software parts. Copilot agentic coding will be used (experimented with) for some parts. Class is divided into three teams of the sub-projects: # Sprengel Museum exhibitions website; # DNB records ‘Sprengel Museum’ # Text and Data Mining: Library catalogue Sprengel Museum Each team carries out the same tasks for their parts to complete a round of data modeling: # Collect data – bottom-up method # Validate the data – top-down method # Presentation of data – Quarto website ‘[https://nfdi4culture.github.io/linked-open-exhibition/ Linked Open Exhibitions]’ Goal: Definition-of-done (DoD) ''NB: Developer speak'' * A documented data model (Table) with diagram (Mermaid, GraphVis, or Draw.io) * Mapping of data model to schemas (Table) * Idea for presentation of data for each sub-section in the prototype and implementation with Publication Manager assistance, e.g., for DNB a chronological list of exhibitions with images. * Documentation of AI LLM use as an assistant, attribution, and comments on good practice * Data provenance and good practice checklist completion * The final result ‘Class Project – Prototyping: Linked Open Exhibitions’ will be made as an institutional deposit with [https://zenodo.org/ Zenodo]. --- ==== Activity #1: Collecting data and bottom up data modeling ==== Confirm, create, or expand existing data models by looking at the source. Each project has a source: * Team #1: Sprengel Museum website, exhibition listings: ** https://www.sprengel-museum.de/ and ** spreadsheet of entered exhibitions CSV GitHub | Spreadsheet TIB Cloud (passworded) ** In prototype: https://nfdi4culture.github.io/linked-open-exhibition/exhibitions.html * Team #2: DNB records of search for ‘Sprengel Museum’: ** https://portal.dnb.de/opac/moveDown?currentResultId=Sprengel+and+Museum%26any&categoryId=books | https://wikibase.wbworkshop.tibwiki.io/ ** CSV https://github.com/NFDI4Culture/linked-open-exhibition/blob/main/catalogues/sprengel_exhibitions.csv ** Images of book covers https://github.com/NFDI4Culture/linked-open-exhibition/tree/main/catalogues/images * Team #3: HsH Library information on catalogues for ‘Sprengel Museum’: ** https://katalog.bib.hs-hannover.de/vufind/Search/Results?lookfor=Sprengel%2BMuseum ** Team 3 have to start from scratch as as yet we don’t have a library record or item in an exhibition data model. Tip: look back at the other models to start building up your data models. '''>>> Add to data model here:''' https://tib.cloud/s/PicTdwCEqCQ6pBp (passworded) ==== OBJECTIVE ==== To ensure that the data model can represent the source. Are there enough entries to describe the things that make up the source. The process is iterative, meaning it keeps on being repeated with improvements and changes being made. ==== TASKS ==== * Edit the purple grey area, thje green are will be edited in the next activity * Review and correct existing information * Add new concepts if the source needs it * Orange areas need filling in. The cells might need editing or added to. * Data types can be found on Property Pages only Items (QIDs) don’t have data types, in <nowiki>https://www.wikidata.org/wiki/Property:P1476</nowiki> under the Label - ''Data type'' * URI is equivelent to URL Tips * Look at other examples on Wikidata: Artists, exhibitions, catalogues, bibliographic, or items in an exhibition. * Use an AI to lookup schema explanations or options. Register with KISSKI to get better AI privacy use. ==== Activity #2: Top down data modeling validation ==== * Team #1: Sprengel Museum website * Team #2: DNB records of search for ‘Sprengel Museum’ * Team #3: HsH Library information on catalogues for ‘Sprengel Museum’ ===== OBJECTIVE ===== Map all concepts ==== TASKS ==== Look up the concept in the different resources and add mapping links, Work time: Open time-slot to review running ‘Tech Stack’ or address any other questions --- ===== Homework ===== * Complete the Bottom-up and Top-down modelling * Team #3: Visit the library and make a digital scan on a copier machine, store as PDF. The scan will be used for text and data mining and the file deleted and destroyed after. We will only be extracting metadata from the scan. * Come along to the next class with ideas and suggestions for what you would like to have displayed from your data and data models in the prototype. [[Category:Wikidata]] jykvclb9hbq3tupclk1flyzlvnmjket 0 328055 2807084 2805591 2026-04-30T04:59:56Z Ckrebs12 3052082 edited my last four modules 2807084 wikitext text/x-wiki == '''<big>Communication Law in Spain</big>''' == ===== Introduction ===== Spain is a parliamentary constitutional monarchy located in southwestern Europe.<ref name=":0">{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1978-31229#:~:text=24%3A%20%23a18%5D-,Art%C3%ADculo%2018,2.|title=Constitución Española (1978)|website=www.constituteproject.org|language=en|access-date=2026-03-02}}</ref><ref name=":36">{{Cite web|url=https://european-union.europa.eu/principles-countries-history/eu-countries/spain_en|title=Spain – EU country {{!}} European Union|website=european-union.europa.eu|language=en|access-date=2026-03-05}}</ref> While the monarch serves as head of state, political power is exercised through a democratic parliamentary system led by a prime minister and the national legislature known as the Cortes Generales. Spain is also a highly decentralized state composed of seventeen Autonomous Communities, each with its own regional government and authority over areas such as culture, language policy, and public broadcasting.<ref name=":0" /> In addition to its domestic institutions, Spain operates within a broader European legal framework as a member of the European Union and a party to international human rights agreements such as the European Convention on Human Rights.<ref name=":36" /> Modern debates over communication law in Spain are also deeply influenced by the country’s twentieth-century history. From 1939 until 1975 Spain was governed by the authoritarian dictatorship of General Francisco Franco, during which the state exercised strict control over political speech and media institutions.<ref>{{Cite web|url=https://www.history.com/articles/francisco-franco|title=Francisco Franco - Biography, Facts & Death|last=Editors|first=HISTORY com|date=2009-11-09|website=HISTORY|language=en|access-date=2026-03-05}}</ref> Following Franco’s death, Spain underwent a democratic transition that culminated in the adoption of the 1978 Constitution, which established modern protections for freedom of expression and democratic pluralism.<ref>{{Cite web|url=https://adst.org/2016/06/spains-post-franco-emergence-dictatorship-democracy/|title=Spain’s Post-Franco Emergence from Dictatorship to Democracy – Association for Diplomatic Studies & Training|language=en-US|access-date=2026-03-05}}</ref> These historical experiences continue to shape contemporary debates over speech, protest, and public memory in Spain. == <big>Sources and Institutions Of Communication Law In Spain</big> == === '''National Sources and Institutions''' === ===== Constitutional Foundations of Communication Law ===== The Spanish Constitution of 1978 is the supreme legal authority governing communication rights in Spain.<ref name=":0" /> Only a few provisions directly address communication, but they shape disputes involving the press, privacy, defamation, surveillance, and protest.<ref name=":0" /> The first major provision is Article 20, which protects freedom of expression and information.<ref name=":0" /> It guarantees freedom of expression, creative and academic freedom, the right to communicate and receive truthful information, and the prohibition of prior censorship.<ref name=":0" /> This is the backbone of Spanish communication law. But Article 20 is not a blank check. Article 20(4) makes clear that expression has limits when it collides with other constitutional rights. <ref name=":0" /> In other words, Spain builds speech protection and speech limits into the same constitutional design. That leads to the second key provision: Article 18, which protects privacy, honor, and the secrecy of communications. Article 18 expressly protects the right to honor, personal and family privacy, personal image, and the secrecy of communications.<ref name=":0" /> These protections frequently arise in modern communication disputes. For example, in Spanish Constitutional Court decision STC 104/1986, the court examined whether a newspaper report accusing a businessman of misconduct violated his constitutional right to honor, emphasizing the need to balance expression with protection of reputation.<ref>{{Cite web|url=https://hj.tribunalconstitucional.es/en/Resolucion/Show/104|title=HJ System - Decision: SENTENCIA 62/1982|website=hj.tribunalconstitucional.es|access-date=2026-03-05}}</ref> Another important principle in the Spanish constitutional system is the protection of the “essential content” of fundamental rights, often referred to as the ''núcleo esencial''. Rooted in Article 10 and Section I on fundamental rights, this principle holds that certain core aspects of rights cannot be undermined by the state.<ref>{{Cite web|url=https://www.lamoncloa.gob.es/lang/en/espana/leyfundamental/paginas/titulo_primero.aspx|title=Part I Fundamental Rights and Duties|website=www.lamoncloa.gob.es|language=en|access-date=2026-03-05}}</ref> The doctrine reflects Spain’s constitutional commitment to human dignity and the free development of personality. In practice, rights may be regulated but not restricted in ways that destroy their core substance. Rights such as expression, life, and physical integrity retain a protected core beyond ordinary political decision-making. An interesting wrinkle in the Spanish Constitution is Article 10(2), often called the international interpretation clause.<ref name=":0" /> It requires that constitutional rights be interpreted in conformity with international human rights treaties ratified by Spain. That strengthens the influence of European and international human-rights standards inside Spain’s own constitutional system. For example, in ''Stern Taulats and Roura Capellera v. Spain'' (2018), the European Court of Human Rights ruled that Spain violated freedom of expression after protesters were convicted for burning photographs of the King during a political demonstration, illustrating how international courts shape constitutional speech protections.<ref>{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22001-181719%22%5D%7D|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-05}}</ref> ===== Regulatory Authorities ===== Spain relies on regulatory authorities to implement and supervise communication law. The National Commission on Markets and Competition (CNMC) oversees telecommunications and audiovisual markets in Spain, with a role that blends sector oversight with competition regulation.<ref>{{Cite web|url=https://www.cnmc.es/|title=Comisión Nacional de los Mercados y la Competencia {{!}} CNMC|website=www.cnmc.es|access-date=2026-03-02}}</ref> The Spanish Data Protection Agency (AEPD) enforces the GDPR and Organic Law 3/2018, and it is one of the main places where “digital rights” become real—through guidance, enforcement, and sanctions.<ref>{{Cite web|url=https://www.aepd.es/|title=Agencia Española de Protección de Datos {{!}} AEPD|website=www.aepd.es|access-date=2026-03-02}}</ref><ref name=":3">{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2018-16673|title=BOE-A-2018-16673 Ley Orgánica 3/2018, de 5 de diciembre, de Protección de Datos Personales y garantía de los derechos digitales.|website=www.boe.es|access-date=2026-03-02}}</ref><ref>{{Cite web|url=https://digital.gob.es/|title=Regulation - 2016/679 - EN - gdpr - EUR-Lex|website=eur-lex.europa.eu|language=en|access-date=2026-03-02}}</ref> The Ministry for Digital Transformation and Public Function plays a coordinating role for national telecommunications and digital policy, including the domestic implementation of EU digital regulation. The ministry also oversees the allocation of radio frequencies, a critical responsibility because the radio spectrum is a limited public resource used by mobile networks, television broadcasting, satellite communications, and other wireless technologies.<ref name=":12">{{Cite web|url=https://digital.gob.es/|title=Portal MTDFP {{!}} Inicio|website=digital.gob.es|access-date=2026-03-02}}</ref><ref name=":9" /> ===== National Legislative Framework ===== Spain does not rely solely on the Constitution and international treaties to regulate communication.<ref name=":0" /><ref name=":5">{{Cite web|url=https://www.echr.coe.int/european-convention-on-human-rights|title=European Convention on Human Rights (ECHR) - EUR-Lex|date=2009-12-01|website=eur-lex.europa.eu|language=en|access-date=2026-03-02}}</ref><ref name=":11" /> Spain can pass national legislation governing communication as long as it stays consistent with superior constitutional and supranational law.<ref name=":0" /><ref name=":4">{{Cite web|url=https://curia.europa.eu/site/jcms/d2_5093/en/the-court-of-justice|title=Court of Justice of the European Union|website=curia|language=en|access-date=2026-03-02}}</ref> The General Audiovisual Communication Law (Law 13/2022) regulates television, radio, and on-demand audiovisual services, including licensing, protection of minors, advertising standards, and media pluralism.<ref name=":1">{{Citation|title=Ley 13/2022, de 7 de julio, General de Comunicación Audiovisual|url=https://www.boe.es/eli/es/l/2022/07/07/13|date=2022-07-08|accessdate=2026-03-02|pages=96114–96220|issue=Ley 13/2022|last=Jefatura del Estado}}</ref> It also functions as Spain’s main implementation of AVMSD requirements.<ref name=":1" /><ref name=":10" /> The General Telecommunications Law (Law 11/2022) regulates electronic communications networks and services, including spectrum allocation and operator licensing.<ref name=":2">{{Cite web|url=https://ppp.worldbank.org/library/general-de-telecomunicaciones-ley-11-2022|title=General de Telecomunicaciones Ley 11/2022|website=PUBLIC-PRIVATE-PARTNERSHIP LEGAL RESOURCE CENTER|language=en|access-date=2026-03-02}}</ref> Under Article 149.1.21 of the Constitution, telecommunications is an exclusive competence of the State.<ref name=":0" /> In other words, national control ensures consistent regulation of telecommunications across Spain’s 17 Autonomous Communities.<ref name=":0" /><ref name=":2" /> On privacy, Spain applies the EU’s General Data Protection Regulation (GDPR) and complements it through Organic Law 3/2018 (LOPDGDD), which regulates data processing and sets out digital rights in domestic law.<ref name=":3" /><ref name=":12" /> This framework includes digital rights such as the “right to erasure” (“right to be forgotten”).<ref name=":3" /><ref name=":12" /> Finally, Organic Law 1/1982 on the Protection of Honor, Privacy, and Personal Image provides civil remedies when freedom of expression conflicts with personal dignity, basically, when speech unlawfully harms reputation or private life.<ref name=":13">{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1982-11196|title=BOE-A-1982-11196 Ley Orgánica 1/1982, de 5 de mayo, de protección civil del derecho al honor, a la intimidad personal y familiar y a la propia imagen.|website=www.boe.es|access-date=2026-03-02}}</ref> This law operationalizes the protections in Article 18 in everyday disputes involving media reporting and personal reputation.<ref name=":0" /><ref name=":13" /> ===== Regional (Autonomous Community) Regulation ===== [[File:Autonomous communities of Spain no names-gl.svg|thumb|'''The 17 Autonomous communities of Spain''']] Spain is a decentralized state composed of 17 Autonomous Communities. While telecommunications remains a national competence under Article 149, Autonomous Communities still influence communication in meaningful ways, especially through public broadcasting and language policy.<ref name=":0" /> Autonomous Communities may create and regulate their own public broadcasting corporations. For example, Catalonia operates the Corporació Catalana de Mitjans Audiovisuals (CCMA)<ref name=":6">{{Cite web|url=https://www.3cat.cat/corporatiu/en/el-grup/|title=The Group - CCMA|last=3Cat|website=3Cat|language=en|access-date=2026-03-02}}</ref> and the Basque Country operates EITB.<ref name=":7">{{Cite web|url=https://www.eitb.eus/en/|title=EITB|website=www.eitb.eus|language=en|access-date=2026-03-02}}</ref> These bodies operate under regional frameworks but still sit under national and EU rules that shape audiovisual services more generally.<ref name=":1" /><ref name=":10" /> Some regions also maintain audiovisual supervisory authorities. Catalonia, for example, has the Consell de l’Audiovisual de Catalunya (CAC), which oversees audiovisual services within the region and has a particular focus on standards tied to language and culture.<ref>{{Cite web|url=https://www.cac.cat/|title=Consell de l'Auidovisual de Catalunya}}</ref> Regional governments also regulate language and cultural policy. Autonomous Communities with co-official languages may adopt measures that promote regional-language media and broadcasting quotas.<ref name=":0" /> These policies shape what audiences actually see and hear day-to-day, but they still must remain consistent with Spain’s constitutional protections and EU standards.<ref name=":0" /><ref name=":8" /> === '''International''' '''Sources and Institutions''' === ===== European Union Law ===== As an EU Member State, Spain is bound by European Union law, including the principle that EU law has primacy in areas where the EU has competence.<ref name=":4" /> EU rules increasingly shape digital communication and audiovisual markets.<ref name=":8">{{Citation|title=Charter of Fundamental Rights of the European Union|url=http://data.europa.eu/eli/treaty/char_2012/oj/eng|date=2012-10-26|accessdate=2026-03-02|language=en}}</ref><ref name=":9">{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2022/2065/oj/eng|title=Regulation - 2022/2065 - EN - DSA - EUR-Lex|website=eur-lex.europa.eu|language=en|access-date=2026-03-02}}</ref><ref name=":10">{{Citation|title=Directive 2010/13/EU of the European Parliament and of the Council of 10 March 2010 on the coordination of certain provisions laid down by law, regulation or administrative action in Member States concerning the provision of audiovisual media services (Audiovisual Media Services Directive) (Codified version) (Text with EEA relevance)|url=http://data.europa.eu/eli/dir/2010/13/oj/eng|date=2010-03-10|accessdate=2026-03-02|volume=095|language=en}}</ref> Article 11 of the EU Charter of Fundamental Rights protects freedom of expression and media pluralism.<ref name=":8" /> When acting within EU law, Spanish authorities must comply with these protections. Two major EU instruments show how direct this influence can be. First, the Digital Services Act (Regulation (EU) 2022/2065) regulates online platforms and intermediary services across the EU, with transparency duties, processes for handling illegal content, and heightened obligations for very large online platforms.<ref name=":9" /> Spain must enforce these rules through its national system. Second, the Audiovisual Media Services Directive (AVMSD) sets EU-wide standards for television and on-demand audiovisual services, including advertising rules, protections for minors, and promotion of European content.<ref name=":10" /> Spain’s General Audiovisual Communication Law (2022) implements these European requirements in national law.<ref name=":1" /> ===== International Obligations ===== Spain is also a party to major international human rights treaties that shape communication law. Spain participates in the World Intellectual Property Organization (WIPO), a specialized agency of the United Nations that administers international systems for protecting intellectual property. WIPO maintains global databases for searching patents, trademarks, and industrial designs across jurisdictions.<ref>{{Cite web|url=https://www.wipo.int/|title=WIPO - World Intellectual Property Organization|website=www.wipo.int|language=en|access-date=2026-03-05}}</ref> For example, the PATENTSCOPE database allows users to search millions of international patent applications filed under the Patent Cooperation Treaty, while the Global Brand Database provides access to trademark records from national and international registries.<ref>{{Cite web|url=https://patentscope.wipo.int/search/en/search.jsf|title=WIPO - Search International and National Patent Collections|website=patentscope.wipo.int|access-date=2026-03-05}}</ref> These tools help prevent conflicting claims and support cross-border protection of intellectual property. Spain is also a party to the International Covenant on Civil and Political Rights (ICCPR), which protects freedom of expression in Article 19.<ref name=":11">{{Cite web|url=https://www.ohchr.org/en/instruments-mechanisms/instruments/international-covenant-civil-and-political-rights|title=International Covenant on Civil and Political Rights|website=OHCHR|language=en|access-date=2026-03-02}}</ref> Because of Article 10(2) of the Spanish Constitution, Spain’s courts must read domestic constitutional rights consistently with these kinds of international commitments.<ref name=":0" /><ref name=":11" /> A key upshot of this layered legal system is that freedom of expression in Spain is not at the mercy of the national political process alone.<ref name=":0" /><ref name=":5" /><ref name=":11" /> Because Spain operates within a broader European and international legal order, attempts to narrow expression face external legal constraints.<ref name=":5" /><ref name=":11" /> This layered system makes it less likely that core expressive freedoms will be reduced. == <big>Freedom of Expression and Dignity in Spain</big> == ===== Constitutional Balance: Expression and Honor ===== Spain protects freedom of expression under Article 20 of the 1978 Constitution, which guarantees the right to express and disseminate ideas and to communicate and receive truthful information, while prohibiting prior censorship.<ref name=":0" /> At the same time, Article 18 protects the right to honor, privacy, and personal image, protections that are further implemented through Organic Law 1/1982 on the Protection of Honor, Privacy, and Personal Image.<ref name=":0" /><ref name=":13" /> The substance of these two provisions often collide, especially because Spanish courts treat them as equally serious constitutional commitments. Unlike systems that treat speech as nearly absolute, Spain’s Constitutional Court uses a balancing approach. When expression conflicts with dignity or reputation, courts weigh the competing rights while ensuring that the essential content (''núcleo esencial'') of each constitutional right is preserved, meaning that neither freedom of expression nor the protection of honor and privacy may be restricted in a way that destroys their core substance.<ref name=":0" /><ref name=":13" /> In practice, this can mean allowing strong criticism of public officials or institutions when it contributes to democratic debate, while still permitting legal remedies when speech crosses into false factual allegations or serious attacks on personal reputation. Spanish constitutional jurisprudence has repeatedly emphasized that freedom of expression has a “preferred position” in democratic debate, especially when speech concerns political issues or public officials.<ref name=":0" /><ref name=":5" /> But that preferred position does not make it untouchable. This framework reflects Spain’s transition to democracy after the Franco dictatorship.<ref>{{Cite web|url=https://www.realinstitutoelcano.org/en/work-document/international-dimensions-of-democratisation-revisiting-the-spanish-case/|title=International dimensions of democratisation: revisiting the Spanish case|last=Powell|first=Charles|website=Elcano Royal Institute|language=en-US|access-date=2026-03-02}}</ref> The 1978 Constitution placed strong emphasis on open public debate as essential to pluralism.<ref name=":0" /> At the same time, dignity is considered a foundational value of the constitutional order. This dual commitment to democratic openness and protection of personal honor defines Spain’s speech doctrine. ===== The Importance of Veracity ===== One distinctive feature of Spanish law is the requirement of “veracity.” Veracity in ethics is the principle of truth-telling, requiring professionals to be honest, transparent, and accurate in all communications to foster trust. Article 20 protects the right to communicate “truthful information.”<ref name=":0" /> Courts do not interpret this to mean that journalists must prove absolute truth.<ref name=":13" /> Instead, they must show that they acted with reasonable diligence in verifying their information. This standard recognizes human limits: reporters and witnesses cannot know “the whole truth.” What matters is whether they checked reliable sources and acted in good faith. If they do, even mistaken reporting may still be protected. If they fail to verify serious factual claims that harm someone’s reputation, liability may follow.<ref name=":13" /> The Constitutional Court has distinguished sharply between opinions and factual statements.<ref name=":13" /> Opinions, especially political opinions, receive strong protection, even when harsh or offensive. Factual allegations that damage someone’s honor are treated differently. In defamation cases, courts examine whether the information contributed to public debate or merely harmed reputation without public interest.<ref name=":13" /> ===== Terrorism, the Monarchy, and Controversial Speech ===== The limits of Spain’s balancing approach become most visible in politically sensitive cases. Following decades of violence by the Basque terrorist group ETA, Spain criminalized the glorification of terrorism and humiliation of victims under Article 578 of the Criminal Code.<ref>{{Cite web|url=https://www.wipo.int/wipolex/en/legislation/details/18760|title=Penal Code (Organic Law No. 10/1995 of November 23, 1995, as amended up to Organic Law No. 2/2019 of March 1, 2019), Spain, WIPO Lex|website=www.wipo.int|language=en|access-date=2026-03-02}}</ref><ref name=":14">{{Cite web|url=https://fibgar.es/en/the-human-rights-committee-urges-spain-to-protect-freedom-of-expression-and-human-rights-defenders/|title=The Human Rights Committee urges Spain to protect freedom of expression and human rights defenders|last=Fibgar|date=2025-08-06|website=FIBGAR|language=en-US|access-date=2026-03-02}}</ref> Supporters argue that these laws protect democratic stability and the dignity of victims. Critics argue that they have sometimes been applied too broadly, including against musicians and social media users.<ref name=":15">{{Cite web|url=https://www.amnesty.org/en/documents/eur41/001/2014/en/|title=Spain: The right to protest under threat|date=2014-04-24|website=Amnesty International|language=en|access-date=2026-03-02}}</ref> In Otegi Mondragón v. Spain (2011), a Basque politician was convicted for referring to the King as the “chief of the torturers.”<ref name=":16">{{Cite web|url=https://hudoc.echr.coe.int/spa?i=001-103951|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-02}}</ref> Spain’s courts upheld the conviction, but the European Court of Human Rights ruled that the conviction violated freedom of expression under Article 10 of the European Convention on Human Rights. The Strasbourg court emphasized that political speech, even when provocative, deserves heightened protection and that public institutions must tolerate stronger criticism.<ref name=":16" /> A similar controversy arose in Stern Taulats and Roura Capellera v. Spain (2018), involving protesters who burned photographs of the King during a political demonstration.<ref name=":17">{{Cite web|url=https://hudoc.echr.coe.int/eng?i=001-181724|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-02}}</ref> Spanish courts treated the act as an insult to the Crown. The European Court again ruled that Spain had violated freedom of expression, finding that the act was symbolic political protest rather than incitement to violence.<ref name=":17" /> Artistic expression has also generated debate. The prosecution of rappers such as Valtonyc for lyrics praising terrorist groups or insulting state institutions sparked international criticism.<ref>{{Cite web|url=https://arisa-project.eu/the-presumption-of-innocence-and-the-media-coverage-of-criminal-cases/|title=The Presumption of Innocence and the Media Coverage of Criminal Cases|last=admin|date=2021-05-13|website=Arisa|language=en-US|access-date=2026-03-02}}</ref><ref name=":18">{{Cite web|url=https://njc.dk/wp-content/uploads/2018/04/Putting-the-chill-in-media-freedom-and-free-speech-.pdf|title=Putting the chill in media freedom and free speech}}</ref>Some observers argued that criminal sanctions risked chilling artistic freedom.<ref name=":18" /> Others defended the prosecutions as necessary to prevent normalization of violence.<ref name=":14" /> These cases reveal a deeper tension in Spain over how far a democracy can go in protecting institutional dignity and social peace without narrowing the space for dissent. ===== Ongoing Debate: Dignity-Centered Democracy ===== Spain’s speech model is often described as dignity-centered. Human dignity is explicitly recognized in Article 10 of the Constitution as a foundational principle of the legal order.<ref name=":0" /> Courts therefore treat attacks on honor, reputation, or institutional integrity as constitutionally significant.<ref name=":0" /><ref name=":13" /> Some scholars argue that this model reflects a mature constitutional democracy that refuses to sacrifice personal dignity in the name of absolute speech.[35][36] They see Spain’s approach as consistent with broader European human rights traditions, where proportionality and balancing are central.<ref name=":5" /><ref name=":16" /> Others argue that criminal penalties for offensive speech, especially in political or artistic contexts, create a chilling effect and discourage open debate.<ref name=":15" /><ref name=":16" /> They point to repeated rulings from the European Court of Human Rights pushing Spain toward stronger protection of political expression.<ref name=":16" /><ref name=":17" /> Spain’s doctrine continues to evolve through judicial dialogue between national courts and European institutions.<ref name=":5" /><ref name=":16" />The result is a system that seeks to protect democratic debate while also preserving the constitutional value of dignity, a balance that remains contested and actively debated. == <big>Spain’s 2015 Citizen Security Law (“Gag Law”)</big> == [[File:Manifestación contra la Ley Mordaza en Madrid 20-12-2014 - 07.jpg|thumb|On December 20, 2014, protesters in Madrid demonstrated against Spain’s new Citizens Security Law, known as the "Gag Law" (Ley Mordaza)]] The Citizen Security Law (Ley Orgánica 4/2015 de protección de la seguridad ciudadana) is a Spanish national law that entered into force on 1 July 2015.<ref name=":19">{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2015-3442|title=BOE-A-2015-3442 Ley Orgánica 4/2015, de 30 de marzo, de protección de la seguridad ciudadana.|website=www.boe.es|access-date=2026-03-02}}</ref> It is widely known in public debate as the “Gag Law” (Ley Mordaza), a nickname given by critics who argue that it discourages protest and limits free expression through financial penalties rather than formal censorship.<ref name=":20">{{Cite web|url=https://www.hrw.org/news/2015/03/09/spain-reject-flawed-public-security-bill|title=Spain: Reject Flawed Public Security Bill {{!}} Human Rights Watch|date=2015-03-09|language=en|access-date=2026-03-02}}</ref><ref name=":21">{{Cite web|url=https://www.amnesty.org/en/latest/news/2015/03/spain-two-pronged-assault-targets-rights-and-freedoms/|title=Spain: Two-pronged assault targets rights and freedoms of Spanish citizens, migrants and refugees|date=2015-03-26|website=Amnesty International|language=en|access-date=2026-03-02}}</ref> The law was introduced by Spain’s government as a modernization of public-order regulations. Officials stated that it was designed to provide clearer rules for police operations, maintain public security, and respond to disruptive protest activity. Supporters emphasize that the law does not establish prior censorship and does not criminalize political opinions as such.<ref name=":19" /> Critics, however, argue that while the Constitution only prohibits prior censorship, the Gag Law creates a system of administrative fines imposed after expression, which can still discourage participation in protests and public criticism. They contend that heavy fines can have a chilling effect, especially on journalists and activists.<ref name=":20" /><ref name=":21" /> ===== Key Provisions of The Citizen Security Law ===== The Citizen Security Law establishes a detailed system of administrative infractions and sanctions. Fines range from several hundred euros to up to €600,000 in the most serious cases.<ref name=":19" /><ref name=":34">{{Cite news|url=https://www.theguardian.com/world/2015/mar/12/spain-security-law-protesters-freedom-expression|title=Spain puts 'gag' on freedom of expression as senate approves security law|last=Kassam|first=Ashifa|date=2015-03-12|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}</ref> Among the most controversial provisions are: * Fines (up to €600) for holding public demonstrations without prior notification, even if peaceful * Fines (up to €600) for protests that deviate from approved routes * Fines (up to €30,000) for protests causing disturbances near Parliament or regional government buildings * Fines (up to €600,000) for unauthorized protests near key infrastructure (airports, nuclear plants, refineries, transport hubs) * Fines (up to €30,000) for obstructing police or officials carrying out evictions or court orders * Fines (up to €30,000) for recording or publishing images of police officers if deemed to endanger their safety or an operation<ref name=":19" /><ref name=":34" /> Human rights organizations have argued that the wording of some provisions is broad and gives authorities significant discretion in enforcement.<ref name=":20" /><ref name=":21" /> ===== International Reaction ===== The “Gag law” was met with strong criticism from international human rights groups even before it entered into force. Human Rights Watch warned that the legislation undermined freedom of assembly and expression by allowing heavy fines for peaceful protest and for recording police conduct.<ref name=":20" /> Amnesty International described the law as a threat to civil liberties and warned that restrictions on filming police could weaken transparency and accountability.<ref name=":21" /> The Committee to Protect Journalists (CPJ) also raised concerns that the law could deter media coverage of demonstrations and police activity.<ref>{{Cite web|url=https://cpj.org/2015/05/why-spains-new-gag-law-is-threat-to-free-flow-of-i/|title=Why Spain's new gag law is threat to free flow of information|last=Blogger|first=Borja Bergareche/CPJ Guest|date=2015-05-01|website=Committee to Protect Journalists|language=en-US|access-date=2026-03-02}}</ref> In addition, United Nations Special Rapporteurs expressed concern that the law’s provisions were overly broad and risked arbitrary enforcement against peaceful protesters.<ref>{{Cite web|url=https://www.ohchr.org/en/press-releases/2015/02/two-legal-reform-projects-undermine-rights-assembly-and-expression-spain-un|title=“Two legal reform projects undermine the rights of assembly and expression in Spain” - UN experts|website=OHCHR|language=en|access-date=2026-03-02}}</ref> ===== Javier Bauluz Case ===== One widely cited case involved Spanish photojournalist Javier Bauluz, a Pulitzer Prize–winning photographer, who was fined €960 under the Citizen Security Law after a confrontation with police while documenting migrant arrivals in the Canary Islands in November 2020.<ref name=":22" /><ref name=":35">{{Cite web|url=https://www.mfrr.eu/spain-fine-against-photographer-underscores-urgent-need-for-reform-of-gag-law/|title=Spain: Fine against photographer underscores urgent need for reform of Gag Law|last=MFRR|date=2022-06-21|website=Media Freedom Rapid Response|language=en-GB|access-date=2026-03-02}}</ref> He had been photographing rescue boats arriving in Arguineguín, where thousands of migrants were being held in conditions later described by a judge as “deplorable.”<ref name=":22" /> Video of the incident shows officers grabbing him and ordering him to leave, and he was later fined for “disrespecting an agent” and “refusing to identify himself,” though he said he had complied and was simply doing his job.<ref name=":22">{{Cite news|url=https://www.theguardian.com/world/2022/jun/14/photographer-capturing-migrant-camp-fined-1000-under-spains-gag-law|title=Photographer capturing migrant camp fined €1,000 under Spain’s ‘gag law’|last=Kassam|first=Ashifa|date=2022-06-14|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}</ref><ref name=":35" /> The fine arrived more than a year later and gave little explanation beyond citing provisions of the law. Bauluz rejected the sanction, arguing that police were limiting press access to prevent journalists from properly documenting the situation.<ref name=":22" /><ref name=":35" /> He criticized the Gag Law for converting disputes into administrative fines imposed directly by authorities rather than matters handled through criminal courts.<ref name=":22" /> The case became a symbol of broader concerns that the law can be used to penalize journalists reporting on police activity. Press freedomorganizations and media groups condemned the fine and called for reform, arguing that the law enables arbitrary sanctions and threatens freedom of expression.<ref name=":22" /><ref name=":35" /> Although Spain’s Constitutional Court upheld most of the law in 2021, critics continue to argue that reform is necessary to bring it in line with international human rights standards.<ref name=":35" /> ===== Constitutional Court Review ===== Spain’s Constitutional Court reviewed the Citizen Security Law following multiple constitutional challenges. In Constitutional Court decision STC 172/2020, the Court upheld most provisions of the law but clarified limits on its application, particularly regarding sanctions for the use or dissemination of images of police officers. The Court emphasized that penalties cannot be applied in ways that effectively restrict legitimate journalistic reporting or public documentation of police activity.<ref name=":19" /><ref name=":37">{{Cite web|url=https://hj.tribunalconstitucional.es/HJ/es/Resolucion/Show/26498|title=Sistema HJ - Resolución: SENTENCIA 172/2020|website=hj.tribunalconstitucional.es|access-date=2026-03-05}}</ref> One of the most controversial aspects of the ruling was the Court’s decision to uphold the provision allowing administrative fines when photographs or videos of police officers are published in ways that could endanger an officer’s safety or interfere with an ongoing operation. The Constitutional Court clarified that the mere act of recording or photographing police officers during public events or demonstrations is not automatically illegal. Instead, sanctions may only be imposed when the dissemination of those images creates a concrete risk to the safety of officers or interferes with a police operation.<ref name=":37" /> For example, publishing images that reveal the identity of undercover officers or expose the location of police units during an active operation could justify sanctions. By contrast, photographing police activity during public demonstrations for journalistic reporting or public accountability generally falls within the protections of freedom of expression.<ref name=":37" /><ref>{{Cite web|url=https://solermartinabogados.com/en/can-i-record-the-police-can-they-force-me-to-erase-the-images-i-have-recorded-of-them/|title=Can I record the police in Spain? rights, limits|date=2025-10-06|language=en-US|access-date=2026-03-05}}</ref> The Court emphasized that enforcement must respect constitutional guarantees of freedom of expression and assembly. However, it did not invalidate the core structure of the law, leaving its administrative sanction framework intact.<ref name=":23">{{Cite web|url=https://www.article19.org/resources/spain-time-to-end-to-repressive-gag-law/|title=Spain: Time to end repressive 'Gag Law'|date=2024-08-20|website=ARTICLE 19|language=en-US|access-date=2026-03-02}}</ref> ===== The Ongoing Debate ===== The Citizen Security Law remains one of the most politically divisive laws in Spain’s contemporary democracy. Supporters argue that the law provides necessary tools to maintain order and protect both police officers and the public. They stress that fines are administrative rather than criminal penalties and are subject to judicial review. From this perspective, the law regulates conduct rather than suppressing political ideas. Critics, by contrast, argue that the law creates a climate of deterrence. Even without criminal prosecution, the risk of substantial fines may discourage citizens from participating in spontaneous demonstrations or from documenting police actions. Civil liberties groups describe this as a “chilling effect” on democratic participation.<ref name=":19" /><ref name=":20" /><ref name=":21" /> Reform efforts have repeatedly emerged in Spain’s national legislature, particularly from left-leaning parties that argue the law should be revised or partially repealed.<ref name=":38">{{Cite web|url=https://www.barrons.com/news/reform-of-spain-s-contested-security-law-fails-9b1f9a5|title=Reform Of Spain's Contested Security Law Fails|last=Presse|first=AFP-Agence France|website=barrons|language=en-us|access-date=2026-03-05}}</ref> These parties contend that provisions related to protest, public demonstrations, and the recording of police activity give authorities too much discretion and risk discouraging political participation. By contrast, many right-leaning parties have defended the law, arguing that it provides necessary tools for maintaining public order and protecting police officers, especially during large demonstrations and periods of political unrest. As a result, proposals to substantially reform the law have often stalled due to political disagreement in parliament.<ref name=":38" /><ref>{{Cite web|url=https://monitor.civicus.org/explore/csos-warn-decision-not-to-reform-gag-law-is-bad-news-for-human-rights-in-spain/|title=CSOs warn decision not to reform “Gag Law” is “bad news for human rights in Spain”|website=Civicus Monitor|language=en|access-date=2026-03-02}}</ref><ref>{{Cite web|url=https://russpain.com/en/news-3/spanish-parliament-stuck-on-security-law-reform-398037/|title=The political scene is heating up: growing disagreements, unexpected pressure and intrigue in parliament|last=Rubio|first=Ricardo|date=2026-02-23|website=RUSSPAIN.COM|language=en-US|access-date=2026-03-05}}</ref> This divide reflects broader political tensions in Spain. Supporters of reform frequently frame the law as a legacy of a more security-focused approach to governance that emerged during periods of economic crisis and protest movements in the 2010s. Opponents of reform argue that weakening the law could undermine the ability of authorities to manage demonstrations and maintain public safety. Because these disagreements map closely onto Spain’s left-right political divide, efforts to significantly change the Citizen Security Law have proven difficult despite ongoing public debate. == <big>Spain’s Historical Memory Act</big> == ===== Historical Background and Democratic Transition ===== [[File:Francisco Franco 1930.jpg|thumb|'''Francisco Franco in 1930, when he was still a rising officer in the Spanish army, years before the Spanish Civil War brought him to power and led to his long dictatorship.''']] Spain’s contemporary debate over historical memory is rooted in the Spanish Civil War (1936–1939) and the subsequent dictatorship of General Francisco Franco, which lasted until 1975.<ref>{{Cite journal|last=Owens|first=Lawrence S.|date=2021|title=Timoteo Mendieta Alcalá and the Pact of Forgetting: trauma analysis of execution victims from a Spanish Civil War mass burial site at Guadalajara, Castilla la Mancha|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC8212665/|journal=Forensic Science International. Synergy|volume=3|pages=100156|doi=10.1016/j.fsisyn.2021.100156|issn=2589-871X|pmc=8212665|pmid=34179739}}</ref> The war divided the country along political, ideological, and religious lines and resulted in widespread repression, imprisonment, and executions.<ref>{{Cite journal|last=Owens|first=Lawrence S.|date=2021|title=Timoteo Mendieta Alcalá and the Pact of Forgetting: trauma analysis of execution victims from a Spanish Civil War mass burial site at Guadalajara, Castilla la Mancha|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC8212665/|journal=Forensic Science International. Synergy|volume=3|pages=100156|doi=10.1016/j.fsisyn.2021.100156|issn=2589-871X|pmc=8212665|pmid=34179739}}</ref><ref name=":24">{{Cite journal|last=Boyd|first=Carolyn P.|date=2008|title=The Politics of History and Memory in Democratic Spain|url=https://www.jstor.org/stable/25098018|journal=The Annals of the American Academy of Political and Social Science|volume=617|pages=133–148|issn=0002-7162}}</ref> After Franco’s victory, the regime promoted an official narrative that framed the conflict as a defense of national unity and Catholic identity.<ref name=":24" /> Public monuments, street names, memorials, and religious symbols commemorating the dictatorship were erected throughout Spain.<ref>{{Cite web|url=https://dash.harvard.edu/server/api/core/bitstreams/e113f9f5-d512-4bfa-bd72-9c150cec2d32/content|title=Historical Memory in Post-Franco Spain: Remembering a Purposely Forgotten Past through Memorialization at the Valle de los Caídos in Cuelgamuros}}</ref> Following Franco’s death in 1975, Spain transitioned to democracy through a negotiated political process often referred to as the “Transition.”<ref>{{Cite web|url=https://api.drum.lib.umd.edu/server/api/core/bitstreams/cda590b3-0ba4-45b8-98c4-4333e42f5ed6/content|title=MEMORY AND RECONCILIATION IN THE SPANISH TRANSITION TO DEMOCRACY: 1975-1982}}</ref> During this period, political leaders adopted what became known as the “Pact of Forgetting” (Pacto del Olvido), an informal political understanding that prioritized reconciliation and democratic stability over reopening Civil War-era grievances.<ref name=":25">{{Cite web|url=https://journals.sagepub.com/action/cookieAbsent|title=Sage Journals: Discover world-class research|website=Sage Journals|language=en|doi=10.1177/026569149702700303|access-date=2026-03-02}}</ref> The 1977 Amnesty Law granted broad amnesty for politically motivated crimes committed during the dictatorship. By the early 2000s, civil society organizations began advocating for greater recognition of victims of Franco-era repression, including efforts to identify mass graves and remove public symbols associated with the dictatorship.<ref>{{Cite web|url=https://scispace.com/pdf/the-return-of-civil-war-ghosts-the-ethnography-of-2j8mponmed.pdf|title=The return of Civil War ghosts The ethnography of exhumations in contemporary Spain}}</ref> Supporters argued that democratic consolidation required public acknowledgment of historical injustices.<ref>{{Cite web|url=https://www.researchgate.net/publication/254084329_Determinants_of_Attitudes_Toward_Transitional_Justice_An_Empirical_Analysis_of_the_Spanish_Case|title=Determinants of Attitudes Toward Transitional Justice: An Empirical Analysis of the Spanish Case}}</ref> In response, Spain enacted Law 52/2007, commonly known as the Historical Memory Act.<ref name=":26">{{Cite web|url=https://reparations.qub.ac.uk/assets/uploads/Ley-52-2007-Spain-EN.pdf|title=Ley 52-2007 Spain EN.docx}}</ref> ===== The 2007 Historical Memory Act ===== Law 52/2007 recognizes and expands rights for individuals who suffered persecution or violence during the Civil War and dictatorship. Its preamble states that it is not the role of the legislator to impose a specific collective memory, but rather to promote democratic values and protect personal and family memory as expressions of democratic citizenship.<ref name=":26" /> At the same time, the law mandates the removal of “shields, insignia, plaques and other objects or commemorative mentions” that exalt the military uprising, Civil War, or repression of the dictatorship from public buildings and spaces.<ref name=":26" /> It also supports efforts to locate and identify victims of repression and provides symbolic recognition to those who suffered under the regime. The Act represents a shift from the earlier policy of institutional silence toward a more active engagement with the legacy of the dictatorship.<ref name=":25" /><ref name=":26" /> The main provisions are: * Official recognition of victims of political, religious, and ideological violence on both sides of the Civil War and under Franco’s rule * Formal condemnation of the Franco regime * Ban on political events at the Valley of the Fallen, where Franco was buried * Removal of public symbols, plaques, statues, and insignia that celebrate the military coup or the dictatorship (with limited exceptions for artistic, architectural, or religious reasons) * Government support for locating, identifying, and exhuming victims buried in mass graves * Spanish citizenship granted to surviving members of the International Brigades without requiring them to give up their original nationality * Declaration that Franco-era trials and laws lacked legitimacy * Temporary changes to nationality rules allowing people who left Spain during the dictatorship — and their descendants — to reclaim Spanish citizenship * Financial and symbolic assistance for victims and their families ===== Supporters’ and Critics’ Perspective ===== Supporters of the Historical Memory Act argue that it reflects a mature democratic commitment to historical justice and human dignity.<ref name=":27">{{Cite news|url=https://www.theguardian.com/world/2022/oct/05/spain-passes-law-to-bring-dignity-to-franco-era-victims|title=Spain passes law to bring ‘justice’ to Franco-era victims|last=Jones|first=Sam|date=2022-10-05|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}</ref><ref name=":28">{{Cite news|url=https://www.nytimes.com/2007/10/28/world/europe/28spain.html|title=Bill in Spanish Parliament Aims to End ‘Amnesia’ About Civil War Victims|last=Burnett|first=Victoria|date=2007-10-28|work=The New York Times|access-date=2026-03-02|language=en-US|issn=0362-4331}}</ref> From this perspective, a constitutional democracy cannot maintain public symbols that glorify authoritarian rule. Removing such symbols is seen not as erasing history, but as ending official state endorsement of a particular political narrative.<ref name=":28" /> Advocates also emphasize the “right to truth” for victims and their families, aligning Spain with broader international human rights standards concerning recognition, memory, and accountability.<ref>{{Cite web|url=https://www.swisspeace.ch/assets/publications/downloads/Gonzalez-Garcia_WorkingPaper_2_2023.pdf|title=The Search for Truth in Spain: Debates Around the Creation of a Truth Commission}}</ref> Reports by United Nations Special Rapporteurs have encouraged Spain to strengthen efforts related to truth, justice, and reparation for victims of Franco-era repression.<ref>{{Cite web|url=https://news.un.org/en/story/2014/02/461222|title=UN expert urges Spain to probe alleged atrocities during 1930's civil war}}</ref> For supporters, the law corrects decades of imbalance in public memory and promotes constitutional values grounded in democracy and human rights.<ref name=":27" /><ref>{{Cite web|url=https://www.amnesty.org/en/documents/eur41/001/2013/en/|title=Spain: Supreme Court overturns ban on full-face veils; AI concerns remain about restrictions on headscarves in schools|date=2013-04-08|website=Amnesty International|language=en|access-date=2026-03-02}}</ref> Critics argue that the Historical Memory Act risks politicizing historical interpretation by privileging one narrative over others.<ref name=":29">{{Cite news|url=https://www.economist.com/europe/2020/09/17/the-spanish-government-proposes-a-new-law-on-history|title=The Spanish government proposes a new law on history|work=The Economist|access-date=2026-03-02|issn=0013-0613}}</ref><ref name=":30">{{Cite news|url=https://www.nytimes.com/2007/10/24/world/europe/24iht-spain.4.8039804.html|title=Spain undergoes wrenching awakening from 'amnesia'|last=Burnett|first=Victoria|date=2007-10-24|work=The New York Times|access-date=2026-03-02|language=en-US|issn=0362-4331}}</ref> Some scholars contend that legislative intervention in historical memory can transform contested historical debate into state-defined orthodoxy.<ref name=":30" /> Opponents also argue that removing monuments may constitute symbolic erasure rather than genuine reconciliation.<ref name=":29" /> They maintain that democratic societies should allow historical interpretation to evolve through open public discourse rather than through statutory mandates.<ref name=":30" /> Much of this debate has centered on the Valle de los Caídos (Valley of the Fallen) memorial complex, one of the most prominent and controversial symbols associated with Spain’s Civil War and the Franco dictatorship. The massive monument, built after the war and located near Madrid, contains a basilica carved into a mountain and a large cross that dominates the surrounding landscape. For decades it served as the burial site of General Francisco Franco as well as thousands of victims from both sides of the Civil War.<ref>{{Cite web|url=https://www.bbc.com/news/world-europe-50164806|title=Franco exhumation: Spanish dictator's remains moved|date=2019-10-24|website=www.bbc.com|language=en-GB|access-date=2026-03-02}}</ref><ref>{{Cite news|url=https://www.theguardian.com/world/2019/oct/24/franco-exhumation-spain-dictator-madrid|title='Spain is fulfilling its duty to itself': Franco's remains exhumed|last=Jones|first=Sam|date=2019-10-24|work=The Guardian|access-date=2026-03-02|language=en-GB|issn=0261-3077}}</ref> Supporters of Spain’s memory laws argue that the site symbolized the continued public prominence of Franco’s regime, while critics argue that the complex represents an important historical monument whose meaning should be debated rather than reshaped through legislation. The controversy intensified when the Spanish government ordered the exhumation of Franco’s remains from the site in 2019, relocating them to a different cemetery.<ref>{{Cite web|url=https://www.lamoncloa.gob.es/lang/en/presidente/news/Paginas/2019/20191024-statement.aspx|title=Institutional statement by Acting President of the Government regarding exhumation of Francisco Franco}}</ref> The government justified the decision as part of a broader democratic memory policy aimed at preventing the memorial from functioning as a place of political homage to the dictatorship. Critics, however, viewed the move as politically motivated and reflective of Spain’s continuing polarization over how the country should confront its past. ===== Ongoing Debate: Truth, Memory, and Democratic Pluralism ===== Spain’s memory laws have become one of the most visible and contested areas of contemporary public debate. The discussion centers on how a democracy should address a painful past and what role the state should play in shaping public memory. In Spain, this debate appears in disputes over monuments, commemorations, public spaces, and the official recognition of victims of the Civil War and Franco’s dictatorship.<ref name=":31">{{Cite web|url=https://www.theartnewspaper.com/2023/06/30/debate-rages-in-spain-over-how-to-rememberor-forgetfranco-dictatorship|title=Debate rages in Spain over how to remember—or forget—Franco's dictatorship|last=Coego|first=Alexandra F.|date=2023-06-30|website=The Art Newspaper - International art news and events|language=en|access-date=2026-03-02}}</ref><ref name=":32">{{Cite web|url=https://jacobin.com/2024/01/spain-memory-law-ghosts-francoism|title=Spain’s Memory Law Hasn’t Banished the Ghosts of Francoism|last=By|website=jacobin.com|language=en-US|access-date=2026-03-02}}</ref> Supporters of the Democratic Memory framework argue that removing Francoist symbols and formally recognizing victims strengthens democracy. They maintain that a constitutional state should not continue to honor an authoritarian regime in public spaces. From this perspective, memory laws do not erase history but instead end state endorsement of dictatorship and affirm the dignity of those who suffered repression.<ref name=":31" /><ref name=":33">{{Cite web|url=https://www.reuters.com/world/spain-pays-tribute-francos-victims-50-years-after-his-death-2025-10-31/|title=Spain pays tribute to Franco's victims 50 years after his death}}</ref> Critics argue that legislating memory can deepen political divisions. Some commentators warn that when the government takes an active role in defining historical meaning, it risks turning complex historical debates into partisan conflicts.<ref name=":32" /><ref>{{Cite web|url=https://jacobin.com/2024/01/spain-memory-law-ghosts-francoism|title=Spain’s Memory Law Hasn’t Banished the Ghosts of Francoism|last=By|website=jacobin.com|language=en-US|access-date=2026-03-02}}</ref> Articles examining Spain’s evolving memory laws describe a society still divided over how to interpret the Civil War and Franco’s legacy, with disagreement over whether these reforms promote justice or contribute to polarization.<ref>{{Cite web|url=https://enrs.eu/article/spanish-controversies-related-to-memory|title=Spanish controversies related to memory|website=ENRS|language=en|access-date=2026-03-02}}</ref> In today’s Spain, historical memory is not only about the past. It remains tied to ongoing debates about national identity, democracy, and constitutional values.<ref name=":32" /><ref name=":33" /> The regulation of collective memory shows how law, history, and public expression intersect in a modern democratic society. == <big>Religious Freedom in Spain</big> == ===== Historical Development ===== Spain’s religious history is defined less by steady liberalization than by recurring struggles over whether religious belief could appear in public at all. Medieval coexistence among Christians, Muslims, and Jews existed, but it never displaced the stronger political impulse toward religious unity enforced through law.<ref name=":39">{{Cite journal|last=Montserrat|first=Daniel B.|date=1995|title=The Constitutional Development of Religious Freedom in Spain: An Historical Analysis|url=https://ir.law.fsu.edu/cgi/viewcontent.cgi?article=1241&context=jtlp|journal=Fla. St. U. J. Transnat’l L. & Pol’y|volume=4|pages=27}}</ref> During the early constitutional period, that impulse was embedded directly into state structures. The Constitution of Cádiz (1812) combined political liberalism with explicit Catholic exclusivity, requiring public officials to swear to defend Catholicism and mandating religious instruction in schools.<ref name=":39" /> Religion was not only protected; it was communicated through state institutions. Later constitutions softened these rules but continued to restrict public expression. Non-Catholic worship was sometimes tolerated, but often confined to private settings, allowing belief without visible organization or expression.<ref name=":39" /> The Spanish Constitution of 1931 marked a sharp shift by restricting the Catholic Church’s institutional role, removing funding, dissolving religious orders, and limiting religious education.<ref name=":40">{{Cite journal|last=Combalia|first=Zolia|last2=Roca|first2=Maria|date=2010|title=Religion and the Secular State of Spain, in Religion and the Secular State|url=https://original.religlaw.org/content/blurb/files/Spain%202014.pdf|journal=(W. Cole Durham, Jr. & Javier Martínez-Torrón eds., 2015)|pages=661}}</ref> Rather than establishing neutrality, this reallocated control over how religion could appear in public institutions. This approach reflected a broader European trend during the early twentieth century, as communist and strongly secular regimes sought to remove religion from public life altogether. In those systems, religious expression was not merely regulated but suppressed—public worship, teaching, and institutional presence were restricted or eliminated and replaced with state-controlled ideological messaging. Spain’s 1931 model did not go as far, but it operated within the same broader movement toward limiting religion’s visibility in public communication and institutional life.<ref name=":40" /> Under Franco, Catholicism was restored as the central organizing force of public life. Religious teaching, symbols, and institutional presence were again integrated into education and law, but limited almost entirely to a single faith.<ref name=":39" /><ref name=":40" /> The 1978 Constitution breaks from both models. Article 16 provides: * “Freedom of ideology, religion and worship of individuals and communities is guaranteed… No one may be compelled to declare his ideology, religion or beliefs… No religion shall have a state character….”<ref>{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1978-31229|title=Constitución Española art. 16 (1978), BOE-A-1978-31229|website=www.boe.es|access-date=2026-04-20}}</ref> This provision protects both private belief and public expression, prevents compelled disclosure, and removes any official state religion, while still allowing cooperation with religious groups. Religion remains visible in public life, but it is no longer directed by the state. ===== Modern Spain: Public Expression, Institutional Practice, and European Context ===== Spain’s modern framework allows religion to operate openly in public life while maintaining formal neutrality. Religious expression appears in education and public institutions as individual and community activity protected by law rather than as state endorsement.<ref name=":40" /> This includes the ability to access religious instruction in public schools when requested and to organize religious communities with legal recognition.<ref name=":40" /> These practices reflect a system where religious expression remains visible in ordinary public settings rather than confined to private belief. This approach aligns with international human rights standards. The Spanish Constitution prohibits discrimination based on religion and guarantees the right to practice religion publicly or privately, consistent with broader protections of religious expression.<ref name=":41">{{Cite web|url=https://www.state.gov/reports/2023-report-on-international-religious-freedom/spain/|title=U.S. Dep’t of State, 2023 Report on International Religious Freedom: Spain (May 1, 2024)|website=United States Department of State|language=en-US|access-date=2026-04-20}}</ref> Spain operates within a broader European human rights framework that applies the margin of appreciation doctrine, allowing individual countries to adopt different approaches to religion in public life.<ref>{{Cite journal|last=Lugato|first=Monica|date=2013|title=The “Margin of Appreciation” and Freedom of Religion|url=https://scholarship.law.stjohns.edu/cgi/viewcontent.cgi?article=1061&context=jcls|journal=J. Cath. Legal Stud.|volume=52|pages=49}}</ref><ref name=":48">{{Cite journal|last=Rayón Ballesteros|first=María Concepción|date=2025|title=Generative AI for Lawyers in Spain: A balanced approach to the legal framework, technical foundations and best practices, combining technological innovation with professional responsibility|url=https://reference-global.com/article/10.2478/law-2025-0003|journal=Complutense University, Law and Business|volume=5|pages=12}}</ref> Under this doctrine, the European Court of Human Rights permits states to balance religious expression and public order according to their own legal traditions.<ref name=":48" /> The contrast with France illustrates this flexibility. France restricts visible religious symbols such as hijabs or large crosses in public schools under Law No. 2004-228 of 15 March 2004,¹⁰ a policy upheld by the European Court of Human Rights in Dogru v. France¹¹ and Kervanci v. France¹². In Spain, by contrast, similar forms of expression are generally permitted, and wearing religious symbols is treated as an individual act rather than a violation of neutrality. Spain’s system also allows religious institutions to participate directly in public education. In Fernández Martínez v. Spain, the European Court of Human Rights upheld Spain’s ability to allow the Catholic Church to control who teaches Catholic religion in public schools.<ref name=":42">{{Cite web|url=https://hudoc.echr.coe.int/eng?i=001-145068|title=Fernández Martínez v. Spain, App. No. 56030/07, Eur. Ct. H.R. (Grand Chamber June 12, 2014)|website=hudoc.echr.coe.int|access-date=2026-04-20}}</ref> The case involved a teacher who lost his position after publicly opposing Church teachings. The Court accepted that religious institutions may define who represents their message in educational settings. Despite formal equality, differences remain in practice, and they are best understood as largely natural rather than artificial. Spain’s legal framework is neutral, but historical and demographic factors shape how religious expression appears. For example, Catholic religious instruction is more widely available in public schools because Catholicism has a larger institutional presence and more students requesting it, not because the law excludes other faiths.<ref name=":40" /><ref name=":42" /> Other groups have the same legal rights but less visible participation due to size and infrastructure. The result is a system in which religion remains active and visible in public life without formal state endorsement. Spain does not remove religion from public space; it regulates how it appears and ensures that participation remains voluntary. == <big>The Right to Be Forgotten</big> == ===== Google Spain and the Transition to the GDPR ===== A seminal case in modern data protection law arose from Spain and reshaped the relationship between privacy and access to information in the digital age. In ''Google Spain SL v. AEPD and Mario Costeja González'', Spain’s Audiencia Nacional asked whether EU data protection law could require a search engine to remove links to lawful, truthful information appearing in name-based searches.<ref name=":43">{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:62012CJ0131|title=Google Spain SL v. Agencia Española de Protección de Datos (AEPD), Case C-131/12 (CJEU 2014).|website=eur-lex.europa.eu|access-date=2026-04-20}}</ref> The dispute stemmed from a 1998 notice in ''La Vanguardia'' announcing the forced sale of Mario Costeja González’s property for unpaid social security debts. Although the publication was lawful, its later digitization made it easily accessible through search engines, effectively reviving a long-resolved matter. Costeja requested removal of the links, and the Spanish Data Protection Agency ordered Google to de-list them while allowing the newspaper to remain online.<ref name=":43" /> The Court of Justice of the European Union held that individuals may request removal of links where the information is “inadequate, irrelevant or no longer relevant,” even if the original publication remains lawful.<ref name=":43" /> It reasoned that search engines act as “data controllers” because they organize and present personal data in a way that significantly affects privacy. The legal harm, therefore, arises not from the original publication, but from the amplified visibility created by search engines.<ref name=":44">{{Cite journal|last=Post|first=Robert C.|date=2017|title=Data Privacy and Dignitary Privacy: Google Spain, the Right to Be Forgotten, and the Construction of the Public Sphere|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2953468|journal=Yale Law School, Public Law Research Paper|volume=598}}</ref> Although ''Google Spain'' was decided under Directive 95/46/EC, that framework has since been repealed and replaced by the General Data Protection Regulation (GDPR).<ref>{{Cite web|url=https://eur-lex.europa.eu/eli/dir/1995/46/oj/eng|title=Directive 95/46/EC arts. 2(a), 2(b).|website=eur-lex.europa.eu|access-date=2026-04-20}}</ref><ref>{{Cite web|url=https://gdpr-info.eu/|title=General Data Protection Regulation (GDPR) – Legal Text|website=General Data Protection Regulation (GDPR)|language=en-US|access-date=2026-04-30}}</ref> The repeal did not eliminate the right recognized in the case. Instead, the principle was codified and strengthened in Article 17 of the GDPR, which establishes a “right to erasure.” Article 17 allows individuals to request deletion of personal data that is no longer necessary, relevant, or lawfully processed, and requires controllers, where feasible, to take reasonable steps to inform other entities processing that data, extending the effect of erasure beyond a single source.<ref name=":45">{{Cite web|url=https://gdpr-info.eu/art-17-gdpr/|title=Art. 17 GDPR – Right to erasure (‘right to be forgotten’)|website=General Data Protection Regulation (GDPR)|language=en-US|access-date=2026-04-30}}</ref> Crucially, Article 17 is grounded in Articles 7 and 8 of the Charter of Fundamental Rights of the European Union. Article 7 guarantees the right to respect for private and family life, while Article 8 establishes a distinct right to the protection of personal data, requiring that such data be processed fairly and subject to independent oversight.<ref>{{Cite web|url=https://fra.europa.eu/en/eu-charter/title/title-ii-freedoms|title=Charter of Fundamental Rights of the European Union arts. 7–8.|website=fra.europa.eu|access-date=2026-04-30}}</ref> In ''Google Spain'', these provisions justified the Court’s conclusion that search results displaying outdated personal information can constitute an ongoing interference with private life and that search engines, as data controllers, must respond to requests for removal.<ref name=":43" /> Unlike the Directive, the GDPR applies directly across all Member States, creating a more uniform and enforceable framework. EU law further provides that references to the repealed Directive are to be read as references to the GDPR, preserving continuity between ''Google Spain'' and the current legal regime.<ref name=":45" /> ===== The Modern Doctrine of the Right to Be Forgotten ===== While ''Google Spain'' established the right to be forgotten, subsequent case law has transformed it into a structured doctrine grounded in Article 17 of the GDPR and the Charter. The modern right is not a mechanism to erase the past, but a balancing framework that evaluates whether continued access to personal information remains justified in light of both privacy and communication interests. Under Article 17, individuals may request erasure where data is no longer necessary, is inaccurate, or is unlawfully processed, but the right is not absolute.<ref name=":45" /> Courts apply a case-by-case balancing test rooted in Articles 7 and 8, weighing the individual’s privacy and data protection rights against the public’s interest in access to information. In practice, this inquiry turns on factors such as accuracy, passage of time, the individual’s role in public life, and whether the information contributes to a matter of legitimate public concern. This balancing directly implicates core communication law values, including freedom of expression, the public’s right to receive information, and the preservation of an accurate public record. In ''Google LLC v. CNIL'', the Court addressed the geographic scope of the doctrine.<ref name=":46">{{Cite web|url=https://infocuria.curia.europa.eu/tabs/document?source=document&docid=218105&doclang=EN|title=Google v. CNIL, Case C-507/17 (CJEU 2019).|website=infocuria.curia.europa.eu|access-date=2026-04-20}}</ref> The French data protection authority had ordered Google to apply de-referencing globally, arguing that limiting removal to European domains would render the right ineffective. The Court rejected that position, holding that EU law does not require global de-referencing. Instead, search engines must ensure effective removal within the European Union, including the use of geo-blocking to prevent users in the EU from accessing de-listed results through non-EU domains. This decision reflects a central communication law concern: allowing global removal would effectively export EU privacy law worldwide, raising serious issues of extraterritorial censorship and conflict with jurisdictions that provide stronger protections for speech.<ref name=":46" /> In ''GC and Others v. CNIL'', the Court refined the balancing test for sensitive categories of information.<ref name=":51">{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:62017CJ0136|title=GC & Others v. Commission Nationale de l’Informatique et des Libertés (CNIL), Case C-136/17, ECLI:EU:C:2019:773 (Ct. Just. Eur. Union Sept. 24, 2019).|website=eur-lex.europa.eu|access-date=2026-04-30}}</ref> The case involved requests to remove links containing data about political affiliations, religious beliefs, and criminal history. The Court held that such data requires heightened protection, but not automatic removal. Instead, search engines must determine whether continued access is “strictly necessary” for the public’s right to information.<ref name=":51" /> For example, information about a politician’s past conduct may remain accessible because it informs democratic decision-making, while similar information about a private individual is more likely to be removed. This standard effectively requires search engines to evaluate whether speech contributes to public discourse, placing them in a quasi-adjudicative role traditionally occupied by courts. In ''TU and RE v. Google LLC'', the Court addressed inaccurate or misleading information.<ref name=":54">{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:62020CJ0460|title=TU & RE v. Google LLC, Case C-460/20, ECLI:EU:C:2022:962 (Ct. Just. Eur. Union Dec. 8, 2022).|website=eur-lex.europa.eu|access-date=2026-04-30}}</ref> The applicants challenged articles criticizing their business practices, arguing that the information was false or distorted. The Court held that individuals need not first obtain a judicial ruling to prove falsity. Instead, if they provide relevant and sufficient evidence that the information is manifestly inaccurate, the search engine must delist it, including associated thumbnail images.<ref name=":54" /> This ruling has significant implications for communication law, as it creates a mechanism similar to defamation law within data protection, allowing individuals to challenge harmful or misleading content without initiating formal litigation while requiring platforms to assess the accuracy of speech. At the same time, the Court has emphasized limits grounded in the public interest. In ''Camera di Commercio di Lecce v. Manni'', the Court rejected a request to remove personal data from a public commercial register documenting a past bankruptcy.<ref name=":55">{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:62015CJ0398|title=Camera di Commercio, Industria, Artigianato e Agricoltura di Lecce v. Salvatore Manni, Case C-398/15, ECLI:EU:C:2017:197 (Ct. Just. Eur. Union Mar. 9, 2017).|website=eur-lex.europa.eu|access-date=2026-04-30}}</ref> It held that such records serve the public interest in legal certainty, transparency, and market reliability.<ref name=":55" /> This reflects a longstanding communication law principle: certain categories of information—particularly official records—retain enduring public value and cannot be erased simply because they are reputationally harmful. Taken together, these cases show that the right to be forgotten is really about whether information should still be easy to find through a name search. If it no longer serves a real public purpose, it can be removed from search results; if it does, it stays. In practice, search engines make that call first, which means they end up deciding what information about a person remains visible online. == <big>Spain in the AI Era</big> == ===== Artificial Intelligence and the Changing Nature of Communication ===== Artificial intelligence is reshaping a core assumption of communication law: that speech and images can be reliably traced to a real speaker. Technologies like deepfakes, AI-generated influencers, and algorithmic content systems now allow communication to circulate without a clear human source, raising new questions about attribution, truth, and accountability. Spain offers a useful lens for how legal systems are responding. Spain does not regulate artificial intelligence through a single, unified statute. Instead, it operates within the framework of the European Union’s Artificial Intelligence Act, which establishes a risk-based regulatory model across Member States.<ref name=":47">{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2024/1689/oj/eng|title=Regulation (EU) 2024/1689 (Artificial Intelligence Act).|website=eur-lex.europa.eu|access-date=2026-04-20}}</ref> High-risk systems, such as those used in employment or public decision-making, must comply with obligations including transparency, human oversight, and safeguards against bias.<ref name=":47" /> Whether generative AI tools fall within these categories depends on how they are used and the extent to which they influence real-world or communicative outcomes. At the national level, Spain is developing a Law on the Good Use and Governance of Artificial Intelligence, which will supplement the EU framework and introduce enforcement mechanisms, including the ability to suspend harmful AI systems.<ref name=":49">{{Cite web|url=https://www.cuatrecasas.com/es/spain/propiedad-intelectual/art/anteproyecto-ley-buen-uso-gobernanza-ia|title=Draft Law on the Good Use and Governance of Artificial Intelligence (Anteproyecto de Ley para el Buen Uso y la Gobernanza de la Inteligencia Artificial) (unpublished draft). (approved by Consejo de Ministros Mar. 11, 2025).|website=Cuatrecasas|language=es|access-date=2026-04-20}}</ref> Spain has also created the Agencia Española de Supervisión de Inteligencia Artificial (AESIA) as its central regulator.<ref>Royal Decree 817/2023 of November 8, establishing a controlled testing environment for artificial intelligence systems (Real Decreto 817/2023, de 8 de noviembre), B.O.E. No. 269, Nov. 10, 2023 (Spain).</ref> Notably, through Royal Decree 817/2023, Spain became the first country in the European Union to implement an AI regulatory sandbox, allowing high-risk systems to be tested under real-world conditions and effectively piloting compliance with the EU AI Act before full enforcement.<ref>General Audiovisual Communication Law 13/2022 of July 7 (Ley 13/2022, de 7 de julio, General de Comunicación Audiovisual), B.O.E. No. 163, July 8, 2022 (Spain).</ref> Spain does not yet have a standalone AI statute, but its existing legal framework, particularly in areas like media, privacy, and commercial regulation, already shapes how artificial intelligence operates in practice. ===== How Spain Is Currently Regulating AI ===== Spain’s audiovisual and media laws are beginning to directly address AI-generated communication. Under Law 13/2022 on Audiovisual Communication, content creators, including high-level influencers, can be treated as audiovisual service providers and are responsible for the content they distribute.<ref>{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2022-11311|title=BOE-A-2022-11311 Ley 13/2022, de 7 de julio, General de Comunicación Audiovisual.|website=www.boe.es|access-date=2026-04-30}}</ref> This matters in the AI context because creators using AI-generated avatars, voices, or deepfake-style content for advertising must ensure that such material is not misleading. For example, an influencer using an AI-generated voice to promote a product without disclosure could face liability for deceptive commercial communication. This emphasis on transparency is reinforced in Spain’s draft law on the Good Use and Governance of Artificial Intelligence, which treats the undisclosed use of deepfakes as a serious regulatory violation in many contexts.<ref name=":49" /> Reputation and dignity are also central concerns. Organic Law 1/1982 protects the right to honor, privacy, and self-image, and its application to AI-generated content is increasingly significant.<ref name=":13" /> Disseminating a non-consensual deepfake, such as placing a person’s likeness into fabricated media, can constitute an illegitimate intromission into that person’s rights, even if the content is artificially generated. Spanish doctrine is increasingly moving toward what scholars describe as algorithmic honor: the idea that harm to reputation can arise from automated systems themselves, regardless of human intent. This aligns with Spanish Supreme Court jurisprudence recognizing that reputational harm caused by automated or data-driven systems may still trigger liability where the effect is injurious.<ref>{{Cite web|url=https://vlex.es/vid/922052595|title=See, e.g., Tribunal Supremo [T.S.] [Supreme Court], Judgment No. 35/2023 (Spain).|website=vLex|language=es|access-date=2026-04-30}}</ref> Closely related is the right to correct false information. Organic Law 2/1984 establishes a traditional right of rectification, allowing individuals to demand correction of inaccurate public statements.<ref>{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1984-7248|title=BOE-A-1984-7248 Ley Orgánica 2/1984, de 26 de marzo, reguladora del derecho de rectificación.|website=www.boe.es|access-date=2026-04-20}}</ref> In the digital era, this concept is reinforced by Spain’s Organic Law 3/2018 on Data Protection and Digital Rights, which includes a modern digital rectification right requiring platforms to address inaccurate or misleading personal data.<ref>{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2018-16673|title=BOE-A-2018-16673 Ley Orgánica 3/2018, de 5 de diciembre, de Protección de Datos Personales y garantía de los derechos digitales.|website=www.boe.es|access-date=2026-04-20}}</ref> In practice, this provides a legal tool against AI-generated falsehoods, such as fabricated biographies or hallucinated statements, by requiring platforms or publishers to correct the record. Spain’s approach also extends into advertising law. Under Law 3/1991 on Unfair Competition, commercial practices must not mislead consumers.<ref>{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1991-628|title=BOE-A-1991-628 Ley 3/1991, de 10 de enero, de Competencia Desleal.|website=www.boe.es|access-date=2026-04-30}}</ref> This applies directly to AI-generated endorsements or testimonials. For instance, if a company deploys an AI-generated persona that appears as specific person to promote a product without disclosure, regulators may treat this as deceptive advertising because it manipulates the audience’s trust in human communication. At a structural level, Spain is also confronting the role of algorithms in shaping communication itself. The Rider Law, enacted through Royal Decree-Law 9/2021, requires companies to disclose the parameters and logic of algorithms that affect workers’ conditions.<ref>{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-2021-7840|title=BOE-A-2021-7840 Real Decreto-ley 9/2021, de 11 de mayo, por el que se modifica el texto refundido de la Ley del Estatuto de los Trabajadores, aprobado por el Real Decreto Legislativo 2/2015, de 23 de octubre, para garantizar los derechos laborales de las personas dedicadas al reparto en el ámbito de plataformas digitales.|website=www.boe.es|access-date=2026-04-20}}</ref> While rooted in labor law, this requirement has clear communicative implications because it forces organizations to explain how algorithmic systems decide and communicate outcomes such as hiring, firing, or task allocation. Spanish courts have upheld this transparency obligation, confirming that algorithmic decision-making can be treated as a form of accountable communication within legal frameworks.<ref>{{Cite web|url=https://hj.tribunalconstitucional.es/en/Resolucion/Show/29590|title=Tribunal Constitucional [T.C.] [Constitutional Court], Judgment No. 80/2023 (Spain).|website=hj.tribunalconstitucional.es|access-date=2026-04-30}}</ref> This logic reaches its clearest expression in the Spanish Supreme Court’s BOSCO decision.<ref>{{Cite web|url=https://vlex.es/vid/1091644276|title=Tribunal Supremo [T.S.] [Supreme Court], Judgment No. 1119/2025 (Spain) (BOSCO case).|website=vLex|language=es|access-date=2026-04-30}}</ref> There, the Court required the government to disclose the logic of an automated system used to determine eligibility for public benefits. From a communication law perspective, the ruling treats algorithmic outputs as a form of state communication. If the government uses an automated system to speak to citizens through decisions, it must also explain that reasoning. Transparency thus becomes a constitutional requirement tied to the public’s right to information. At the same time, BOSCO exposes a deeper constitutional tension. The right of access to information under Article 105(b) of the Spanish Constitution may conflict with protections for intellectual property and trade secrets under Article 33.<ref>{{Cite web|url=https://www.boe.es/buscar/pdf/1978/BOE-A-1978-40001-consolidado.pdf|title=Constitución Española, B.O.E. No. 311, Dec. 29, 1978, arts. 33, 105(b) (Spain).}}</ref> The Court’s reasoning suggests that, at least where fundamental rights are implicated, public communicative accountability can outweigh private commercial secrecy. This marks a significant shift in how communication law interacts with technology, as the logic behind speech itself may become subject to disclosure. Across these areas, a common theme emerges. Spain is not treating AI as a separate legal problem requiring entirely new doctrines. Instead, it is adapting existing communication law principles, including truthfulness, transparency, dignity, and accountability, to new technological conditions. The result is a framework in which AI-generated communication is regulated not by its novelty, but by its effects on the public sphere and on individual rights. This remains a rapidly developing landscape as Spain continues to refine its approach alongside evolving European standards. == <big>The Right to One’s Own Image in Spain</big> == ===== The Legal Doctrine of the Right to One’s Own Image ===== Spanish law protects the “right to one’s own image” as a distinct legal interest that governs how a person’s identity—through image, voice, or other identifying features—may be used by others, particularly in media and commercial contexts. This right is codified in Organic Law 1/1982, which provides civil remedies against “illegitimate interference.”<ref name=":50">{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1982-11196|title=BOE-A-1982-11196 Ley Orgánica 1/1982, de 5 de mayo, de protección civil del derecho al honor, a la intimidad personal y familiar y a la propia imagen.|website=www.boe.es|access-date=2026-04-20}}</ref> Article 7.5 treats as unlawful the capture or publication of a person’s image in private contexts, while Article 7.6 prohibits the use of a person’s name, voice, or image for advertising, commercial, or analogous purposes.<ref name=":56">{{Cite web|url=https://www.boe.es/buscar/act.php?id=BOE-A-1982-11196|title=BOE-A-1982-11196 Ley Orgánica 1/1982, de 5 de mayo, de protección civil del derecho al honor, a la intimidad personal y familiar y a la propia imagen. arts. 7.5–7.6.|website=www.boe.es|access-date=2026-04-30}}</ref> Unlike copyright law, which protects creative works, the right to one’s own image protects the individual as the subject of representation. Even where a photograph or video is lawfully owned by a third party, the person depicted retains control over how their likeness is used.<ref name=":57">{{Cite journal|last=Barrnett|first=Stephen R.|date=1999|title=The Right to One's Own Image': Publicity and Privacy Rights in the United States and Spain|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=224628|journal=Am. J. Comp. L.|volume=47|pages=555}}</ref> This distinction is especially important in communication law, where images circulate through media, advertising, and digital platforms independently of the underlying work. Although commonly framed as a visual right, Spanish law explicitly extends protection beyond appearance to other identifying attributes. Article 7.6 includes the use of a person’s “name, voice, or image,” reflecting a broader concern with recognizable identity rather than strictly visual likeness.<ref name=":56" /> While the right originates in Article 18.1 of the Spanish Constitution, which guarantees protection of honor, privacy, and one’s own image, Spanish courts have developed it as an autonomous legal doctrine. Liability does not depend on falsity, reputational harm, or physical intrusion, but instead on the unauthorized use of identifiable personal attributes.<ref>{{Cite web|url=https://www.boe.es/buscar/pdf/1978/BOE-A-1978-40001-consolidado.pdf|title=Constitución Española art. 18.1 (1978) (Spain).}}</ref> In STS 60/1998, the Supreme Court clarified that the key inquiry is whether a person can be recognized, even if not with perfect clarity.<ref>{{Cite web|url=https://vlex.es/vid/proteccion-fundamentales-imagen-as-17745878|title=Tribunal Supremo [T.S.] [Supreme Court], Sala Primera, Jan. 30, 1998 (RJ 1998/358) (Spain).|website=vLex|language=es|access-date=2026-04-30}}</ref> This means that partial or stylized representations, such as silhouettes, blurred images, or distinctive features, may still trigger protection if identification is possible. At the same time, the right operates within a structured set of limits. Article 8 of Organic Law 1/1982 provides exceptions where competing interests prevail, including where there is a “predominant and relevant” cultural or informational interest or where images of public figures are captured in public settings.<ref name=":50" /> Rather than applying rigid categories, courts evaluate whether a particular use is justified in light of its contribution to public discourse or cultural expression.<ref>{{Cite web|url=https://vlex.es/vid/despido-improcedente-despidos-103-3-17760168|title=Tribunal Supremo [T.S.], Sala Primera, Oct. 7, 1996 (RJ 1996/7058) (Spain).|website=vLex|language=es|access-date=2026-04-30}}</ref> ===== From Control to Commerce: How Image Rights Are Monetized ===== These principles become especially concrete in contexts like sports, where image and identity are inseparable from commercial value. Athletes’ likenesses are routinely used by clubs, sponsors, and media, but Spanish law maintains that control over that use remains with the individual.<ref name=":52">{{Cite web|url=https://business-school.laliga.com/en/news/image-rights-in-football|title=Image Rights in Sports {{!}} LaLiga Business School|website=Liga de Fútbol Profesional|language=en|access-date=2026-04-20}}</ref> A footballer, for example, may authorize a club to use his image for team promotions, while separately licensing endorsement rights to a brand. Spanish law structures this through a dual framework: * Negative right: the ability to block unauthorized uses * Positive right: the ability to license and commercially exploit one’s image This explains why contracts in professional sports carefully define scope, duration, and purpose of any assignment.<ref name=":57" /> Crucially, Spanish law adopts a broad understanding of “commercial” use. This broader conception is illustrated by STS 816/1996.<ref name=":53">{{Cite web|url=https://vlex.es/vid/intimidad-imagen-reproduccion-autorizada-17742790|title=STS 816-1996, 7 de Octubre de 1996|website=vLex|language=es|access-date=2026-04-20}}</ref> There, the City of Madrid used photographs of identifiable individuals in a public-awareness campaign promoting respect for the elderly. The Supreme Court held that the use was “publicitario” even without profit, because it relied on identifiable persons to convey its message.<ref name=":53" /> The Court rejected the defense under Article 8.1, emphasizing that the campaign did not require the use of specific individuals’ images to achieve its purpose. By contrast, in STS 21 December 1994, the Court allowed the reuse of a performer’s image to promote a revival of a traditional musical production.<ref>{{Cite web|url=https://vlex.es/vid/202673271|title=Tribunal Supremo [T.S.], Sala Primera, Dec. 21, 1994 (RJ 1994/9775) (Spain).|website=vLex|language=es|access-date=2026-04-30}}</ref> The distinguishing factor was context: the image directly related to the cultural work being promoted, and its use contributed to preserving a recognized artistic tradition. Together, these cases show that Spanish courts focus less on formal categories and more on whether the use is necessary and proportionate to its asserted purpose. Spanish law also extends this protection to voice. In the Tom Waits case (Juzgado de Primera Instancia de Barcelona, 2006), an advertising agency hired a performer to imitate Waits’s distinctive voice after he refused to participate in a commercial. The court held this unlawful, reasoning that imitating a recognizable voice for commercial purposes exploits a person’s identity and misleads the public into believing the individual endorsed the product.<ref>{{Cite journal|last=PONTE|first=LUCILLE M.|date=Winter 2009|title=PRESERVING CREATIVITY FROM ENDLESS DIGITAL EXPLOITATION: HAS THE TIME COME FOR THE NEW CONCEPT OF COPYRIGHT DILUTION?|url=https://www.bu.edu/law/journals-archive/scitech/volume151/documents/ponte_web.pdf|journal=B.U. J. SCI. & TECH. L|volume=Vol. 15.1}}</ref> ===== From Exposure to Use: When the Public Can Reproduce an Image ===== A central question in this doctrine is how far public visibility allows others to reproduce a person’s image. Spanish law recognizes a strong interest in freedom of information, particularly where images contribute to reporting on matters of public concern. In STS 28 December 1996, a newspaper published a photograph of a criminal defendant leaving court. The Supreme Court held the publication lawful because it related to a matter of public interest and contributed to informing the public about judicial proceedings.<ref>{{Cite web|url=https://vlex.es/vid/202746407|title=Tribunal Supremo [T.S.], Sala Primera, Dec. 28, 1996 (RJ 1996/9510) (Spain).|website=vLex|language=es|access-date=2026-04-30}}</ref> The fact that the image was taken in a public setting reinforced this conclusion. However, public exposure does not eliminate the need for consent. This principle becomes especially important in the digital context. In STS 91/2017, a newspaper used a photograph taken from a victim’s Facebook profile when reporting a violent incident.<ref name=":58">{{Cite web|url=https://vlex.es/vid/667177509|title=Tribunal Supremo [T.S.], Sala Primera, Feb. 15, 2017 (RJ 2017/91) (Spain).|website=vLex|language=es|access-date=2026-04-30}}</ref> The Court held that this violated the right to one’s image, emphasizing that making a photograph accessible online does not amount to consent for its reuse. Consent must be specific to each use and cannot be inferred from general availability.<ref name=":58" /> The limits of permissible use become even clearer in cases involving dignity and suffering. In STC 231/1988, the Constitutional Court held that distributing footage of a bullfighter dying after being gored violated the privacy rights of his widow.<ref>{{Cite web|url=https://www.boe.es/buscar/doc.php?id=BOE-T-1988-29203#:~:text=2.%20Do%C3%B1a%20Isabel%20Pantoja%20Mart%C3%ADn,%20ahora%20recurrente,derecho%20a%20la%20intimidad%20y%20a%20la|title=BOE-T-1988-29203 Sala Segunda. Sentencia 231/1988, de 2 de diciembre. Recurso de amparo 1.247/1986. Contra Sentencia de la Sala Primera del Tribunal Supremo que anula la dictada en apelación por la Audiencia Territorial de Madrid, en autos sobre vulneración del derecho a la intimidad. Voto particular.|website=www.boe.es|access-date=2026-04-30}}</ref> Although the event occurred in a public arena, the Court concluded that the dissemination of images capturing extreme distress crossed the boundary of acceptable informational use. As of now, the Spanish right to one’s own image is best understood as a doctrine of controlled visibility. It protects an individual’s authority over how they are represented, even in public-facing contexts such as media, sports, and digital platforms. While the law accommodates competing interests, such as news reporting, cultural expression, and satire, it consistently resists the idea that visibility alone permits unrestricted use. In an environment where images circulate rapidly and widely, this framework ensures that identity remains anchored in the individual rather than absorbed into the commercial or informational systems that reproduce it. [[Category:Communication in Europe|Law in Spain]] [[Category:Law in Europe]] [[Category:Spain]] mgvq3pshjkourtqj19yjxujh4whrv7l 2 328661 2807088 2806534 2026-04-30T07:49:12Z ThinkingScience 3061446 /* April 20th experiment, "AI Decisions, sure. AI-generation NEVER" */ as planned...I might do this today. Less talk more work 2807088 wikitext text/x-wiki == Main focus: my "idea" == * This is my [[Draft:The Neurodiversity-inspired Idea]]. There goes the "main effort" based on my other smaller effort in various places and also by using the methodology I one day hope I will make. * [[User:ThinkingScience/ND_Inspired_Idea_Notebook|Daily Diary of ND Inspired Idea]] * These are my course notes: [[User:ThinkingScience/Draftspace/Coursera]] == Taking responsibility for famous people or people to focus on in [[Draft:The Neurodiversity-inspired Idea]] == I need to take responsibility for the choices I make. If any of my choices resulted in a harm to a real person I am responsible whether I agree or not to any blame being put on me. This section may be moved to a sub-page if I think it starts getting too "cluttered" and later into more sub-pages if the list just grows and grows. === T === Taylor Swift. Why I chose Taylor Swift. I have watched interviews with her before. She is an interesting person to me. I discovered she is open about her creation process. I value that in human beings and that includes people I meet offline, in the "real world" but it will be a challenge for me to make video notes that are "Do no harm". I may be "way in over my head". Please help me if you think I'm doing something wrong. She has a dedicated follower base which may have a large influence. Maybe I'll suffer for this but keeping my "idea" locked inside "my head" I think will cause greater harm to me than good. I know what risk I am taking...or probably not but I gotta move forward or try to. Perhaps my fears are greater than real risks in reality but who knows? == April 20th experiment, "AI Decisions, sure. AI-generation NEVER" == Starting today on April 20th after 08:46 UTC Time(I got UTC time on this computer where I'm so far only using this account), I'll begin by editing Wikiversity resources by being more encouraged by "yeah, do that" comments by Large Language Models. Nothing of it will be "AI-generated" but the decisions I take: the reason for the decisions I take may be because of "AI-generation" but of course I will try to stay away from clear stupidity like if the AI-generation says "jump off a cliff". An extreme example, but I wanted to make a point that I won't take any decision and I will question the "AI/LLM" if it suggests something that to me sounds insane. If you see anything weird please comment on my talk page after you've reverted my edits. When this experiment ends, I don't have a plan for that yet. User input might help. This is where I make notes of decisions that may motivate me to do edits in places. It should include both inputs and outputs and what kind of "version" of "AI"s/LLMs I'm using: * [[/April 20th Experiment Notes|"AI Notes" for motivation purposes in this "experiment"]] == Coursera schedule and notes == Today April 16, 2026 my contributions contain a lot of spelling mistakes. They may be present other days too. You'll probably spot spelling mistakes all over. My studying schedule as I've understood it so far(studying with my mother): This schedule is not reliable(cause my studying partner keeps changing the time, which is not necessarily bad): UTC TIME: 07:30 - 09:30 (2 hours a day, 6 hours a week) * Monday * Thursday * Saturday == I'm studying on Coursera and about their Terms of Use == '''Nothing here is legal advice'''. This is very important. Nothing in this "Wikisection" constitutes legal advice! Please don't blindly follow my advice and if someone copies some parts of this text without providing context then they are responsible for what they share! If you have been tricked by scammers that's sad but I am NOT responsible for illegal activities. * web.archive.org/web/20260325233813/https://www.coursera.org/about/terms "When you create your Coursera account, and when you subsequently use certain features, you must provide us with accurate and complete information, and you agree to update your information to keep it accurate and complete." My interpretation of that is that on Coursera I have to provide a real name. There is a field for "Full name"(retrieved 2026-04-09 UTC YYYY-MM-DD). How does that correspond to these terms? It doesn't say "Real name" but even if it did, what if I choose a name for myself and I'd like to call myself ThinkingScience? Is it still accurate? They don't specify what I actually have to do, just based on my quote. It would be nice for me and other Coursera learners to know what is true. Is the privacy on Wikiversity better? I'd say it is because on Coursera we are forced to provide an email address to create an account. We are not forced to do that on Wikiversity, Wikidata etc. == notes about this account == This account is an alternative account on a computer I don't trust. It should never be allowed to vote and if it does please block this account. It's an alt of [[User:Dekatriofovia]] which unfortunately I have to prove right now despite me being in a hurry...so I'll edit my account at Dekatriofovia at the same time almost and publish at the same time...so you know it's me. The reason for this account is it's on a computer with a bigger screen so I can more easily read books and documents. == a thing I did not regret(modified section title) == This may be blathering but it ends with another Wikilink where I will pass my "idea" through '''Wikiversity:Research ethics''' and through anything else that might be required before anything enters Draft space. The "idea" is "'''The Neurodiversity-inspired idea'''". [[Protoscience]] was an interesting read. I think it will be calming for me if my idea is proven to be pseudoscience cause I can stop worrying about it and leave it behind me. "The Neurodiversity-inspired idea"(in lack for a better name, for now) will not be published in main space, only in draft space. [[Wikiversity:Original research]] made me think "I may be way over my head" (though I stumbled around a bit due to not knowing English at an advanced enough level...this parenthesis is about some unimportant trivia). I'm gonna place everything regarding "The Neurodiversity-inspired idea" into draft space and pass it through '''Wikiversity:Research ethics'''(sorry for repeating myself) and anything else I can find and also ask the community here on Wikiversity what else to place it through. I thought I was gonna create '''User:ThinkingScience/The Neurodiversity-inspired Idea'''(but turns out I was encouraged to create it in Draft: space ... (this paragraph has been modified. Edit history might keep the original). Here are my notes again which I wanted to link to [[User:ThinkingScience/ND Inspired Idea Notebook]] '''Draft:The Neurodiversity-inspired Idea''' that probably is in line with "be bold". === It happened, a small burden has been lifted === I posted to the [[Wikiversity:Colloquium]] https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&oldid=2805080 Thing may be archive in the future. I've lost many things that way.(but also re-discovered many things that landed in the archive that I had posted too!) One week. One small burden lifted. It was the only way forward. I may have been driven insane otherwise or this is just a very bad day I'm having. Full of things that "real life" is demanding of me. More specifically, this is what I posted [[Wikiversity:Colloquium#Advice_needed:_A_Neurodiversity-inspired_Idea/observation]] ptqgz02lkd8iw1e6onkmu0wqlc12uy5 2807090 2807088 2026-04-30T08:32:25Z ThinkingScience 3061446 /* notes about this account */ just doubling down. This is important 2807090 wikitext text/x-wiki == Main focus: my "idea" == * This is my [[Draft:The Neurodiversity-inspired Idea]]. There goes the "main effort" based on my other smaller effort in various places and also by using the methodology I one day hope I will make. * [[User:ThinkingScience/ND_Inspired_Idea_Notebook|Daily Diary of ND Inspired Idea]] * These are my course notes: [[User:ThinkingScience/Draftspace/Coursera]] == Taking responsibility for famous people or people to focus on in [[Draft:The Neurodiversity-inspired Idea]] == I need to take responsibility for the choices I make. If any of my choices resulted in a harm to a real person I am responsible whether I agree or not to any blame being put on me. This section may be moved to a sub-page if I think it starts getting too "cluttered" and later into more sub-pages if the list just grows and grows. === T === Taylor Swift. Why I chose Taylor Swift. I have watched interviews with her before. She is an interesting person to me. I discovered she is open about her creation process. I value that in human beings and that includes people I meet offline, in the "real world" but it will be a challenge for me to make video notes that are "Do no harm". I may be "way in over my head". Please help me if you think I'm doing something wrong. She has a dedicated follower base which may have a large influence. Maybe I'll suffer for this but keeping my "idea" locked inside "my head" I think will cause greater harm to me than good. I know what risk I am taking...or probably not but I gotta move forward or try to. Perhaps my fears are greater than real risks in reality but who knows? == April 20th experiment, "AI Decisions, sure. AI-generation NEVER" == Starting today on April 20th after 08:46 UTC Time(I got UTC time on this computer where I'm so far only using this account), I'll begin by editing Wikiversity resources by being more encouraged by "yeah, do that" comments by Large Language Models. Nothing of it will be "AI-generated" but the decisions I take: the reason for the decisions I take may be because of "AI-generation" but of course I will try to stay away from clear stupidity like if the AI-generation says "jump off a cliff". An extreme example, but I wanted to make a point that I won't take any decision and I will question the "AI/LLM" if it suggests something that to me sounds insane. If you see anything weird please comment on my talk page after you've reverted my edits. When this experiment ends, I don't have a plan for that yet. User input might help. This is where I make notes of decisions that may motivate me to do edits in places. It should include both inputs and outputs and what kind of "version" of "AI"s/LLMs I'm using: * [[/April 20th Experiment Notes|"AI Notes" for motivation purposes in this "experiment"]] == Coursera schedule and notes == Today April 16, 2026 my contributions contain a lot of spelling mistakes. They may be present other days too. You'll probably spot spelling mistakes all over. My studying schedule as I've understood it so far(studying with my mother): This schedule is not reliable(cause my studying partner keeps changing the time, which is not necessarily bad): UTC TIME: 07:30 - 09:30 (2 hours a day, 6 hours a week) * Monday * Thursday * Saturday == I'm studying on Coursera and about their Terms of Use == '''Nothing here is legal advice'''. This is very important. Nothing in this "Wikisection" constitutes legal advice! Please don't blindly follow my advice and if someone copies some parts of this text without providing context then they are responsible for what they share! If you have been tricked by scammers that's sad but I am NOT responsible for illegal activities. * web.archive.org/web/20260325233813/https://www.coursera.org/about/terms "When you create your Coursera account, and when you subsequently use certain features, you must provide us with accurate and complete information, and you agree to update your information to keep it accurate and complete." My interpretation of that is that on Coursera I have to provide a real name. There is a field for "Full name"(retrieved 2026-04-09 UTC YYYY-MM-DD). How does that correspond to these terms? It doesn't say "Real name" but even if it did, what if I choose a name for myself and I'd like to call myself ThinkingScience? Is it still accurate? They don't specify what I actually have to do, just based on my quote. It would be nice for me and other Coursera learners to know what is true. Is the privacy on Wikiversity better? I'd say it is because on Coursera we are forced to provide an email address to create an account. We are not forced to do that on Wikiversity, Wikidata etc. == notes about this account == This account is an alternative account on a computer I don't trust. It should never be allowed to vote and if it does please block this account. Doubling down on this today at 2026-04-30!(intent unchanged) It's an alt of [[User:Dekatriofovia]] which unfortunately I have to prove right now despite me being in a hurry...so I'll edit my account at Dekatriofovia at the same time almost and publish at the same time...so you know it's me. The reason for this account is it's on a computer with a bigger screen so I can more easily read books and documents. == a thing I did not regret(modified section title) == This may be blathering but it ends with another Wikilink where I will pass my "idea" through '''Wikiversity:Research ethics''' and through anything else that might be required before anything enters Draft space. The "idea" is "'''The Neurodiversity-inspired idea'''". [[Protoscience]] was an interesting read. I think it will be calming for me if my idea is proven to be pseudoscience cause I can stop worrying about it and leave it behind me. "The Neurodiversity-inspired idea"(in lack for a better name, for now) will not be published in main space, only in draft space. [[Wikiversity:Original research]] made me think "I may be way over my head" (though I stumbled around a bit due to not knowing English at an advanced enough level...this parenthesis is about some unimportant trivia). I'm gonna place everything regarding "The Neurodiversity-inspired idea" into draft space and pass it through '''Wikiversity:Research ethics'''(sorry for repeating myself) and anything else I can find and also ask the community here on Wikiversity what else to place it through. I thought I was gonna create '''User:ThinkingScience/The Neurodiversity-inspired Idea'''(but turns out I was encouraged to create it in Draft: space ... (this paragraph has been modified. Edit history might keep the original). Here are my notes again which I wanted to link to [[User:ThinkingScience/ND Inspired Idea Notebook]] '''Draft:The Neurodiversity-inspired Idea''' that probably is in line with "be bold". === It happened, a small burden has been lifted === I posted to the [[Wikiversity:Colloquium]] https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&oldid=2805080 Thing may be archive in the future. I've lost many things that way.(but also re-discovered many things that landed in the archive that I had posted too!) One week. One small burden lifted. It was the only way forward. I may have been driven insane otherwise or this is just a very bad day I'm having. Full of things that "real life" is demanding of me. More specifically, this is what I posted [[Wikiversity:Colloquium#Advice_needed:_A_Neurodiversity-inspired_Idea/observation]] hur5ya56x5cimagek7dey6ngel0afst 0 328960 2807037 2806430 2026-04-29T17:18:30Z ~2026-26250-59 3069324 2807037 wikitext text/x-wiki {{Article info | first1 = Julia | last1 = Rogushina | orcid1 = 0000-0001-7958-2557 | affiliation1 = Institute for Digitalisation of Education of the NAES of Ukraine, Kyiv, UA | correspondence1 = {{nospam|adamandraka2010|gmail.com}} | first2 = Anatoly | last2 = Gladun | orcid2 = 0000-0002-4133-8169 | affiliation2 = Institute of Information Technologies And Systems of the NAS of Ukraine, Kyiv, UA | correspondence2 = {{nospam|glanat|yahoo.com}} | first3 = Serhii | last3 = Pryima | orcid3 = 0000-0002-2654-5610 | affiliation3 = Dmytro Motornyi Tavria State Agrotechnological University, Melitopol, Zaporizhzhia, UA | correspondence3 = {{nospam|pryima.serhii|tsatu.edu.ua}} | first4 = Olena | last4 = Anishchenko | orcid4 = 0000-0002-6145-2321 | affiliation4 = Institute of Pedagogical and Adult Education of NAES of Ukraine, Кyiv, UA | correspondence4 = {{nospam|evaler58|ukr.net}} | w1 = | journal = WikiJournal of Humanities | license = | abstract = | submitted = 2025-06-24 }} {{CTA button|cta_link=https://en.wikiversity.org/wiki/WikiJournal_of_Humanities/Proceedings_of_the_1st_International_Scientific_and_Practical_Conference_Wikipedia_and_Wikimedia_projects_in_the_focus_of_scientific_research|cta_text=← Back to Conference Proceedings main page}} '''Abstract''' The publication explores the potential of wiki encyclopedias in addressing the challenge of constructing thesauri for educational courses. It highlights the advantages of utilizing semantic elements from these wiki platforms in the professional development of adult educators. ''Keywords:'' wikiencyclopedias, educational course thesauri, professionalization andragogue. У публікації розкрито потенціал використання енциклопедій на основі вікітехнології для побудови тезаурусів навчальних курсів. Визначено переваги використання семантичних елементів цих вікіенциклопедій у професіоналізації андрагогів. ''Ключові слова:'' вікіенциклопедії, SemanticMediaWiki, тезаурус навчального курсу. idgat9civ9crfhgxi0r5rycfoe2pudy 2 329164 2807028 2806799 2026-04-29T17:03:14Z Atcovi 276019 organizing 2807028 wikitext text/x-wiki == Subpages == * [[User:Atcovi/OGM & Suicide/Papers]] * [[User:Atcovi/OGM & Suicide/The Paper]] * [[User:Atcovi/OGM & Suicide/Poster Build]] kl5idm1yhlb0h8dlpcmx1ygqufph5ym 2 329177 2806987 2806968 2026-04-29T12:17:24Z ThinkingScience 3061446 /* Google's "AI Mode" */ one word too much 2806987 wikitext text/x-wiki Template Links: * {{tl|Draft}} * {{tl|underconstruction}} '''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].''' == "Diary" == == April 18, 2026 == A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a "diary" that will be only regarding the project. Turned "me language" into expressing that everyone is welcome, that I don't "own" [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says "I did this" "I did that" should be gone. I think this was an improvement of some sort. Plan for next edits on the draft page: Add an <nowiki>" == Old Methodology needing updating == "</nowiki> where I will add old methodologies where I had not planned ahead too good and the "Do no harm" I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago. Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future? == April 20, 2026 == Why does it seem like I'm the only one using the word "methodology"? Did the [[Wikiversity:Research ethics]] mention it? * I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now. == April 21, 2026 == I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the "Do no harm". Considering I have almost not developed anything but I still gotta work on this...to do... == April 22, 2026 == I am yearning / looking forward to working on video notes in a "Do no harm" way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different. == April 27, 2026 == I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more. Today I woke up being inspired by that: * Is there a hypothesis or a number of hypotheses related to the idea? ** Can this idea be proven false? Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity. Deadlines etc. can help me keep moving... What a hypothesis is: "I predict this will happen" and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect. [[Operationalization]] is then also needed to make "ambiguous ideas" measurable. My father remembers that I "wanted to save the world" but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea... == April 29, 2026 == Just made my father aware of this "idea" that so far doesn't have a published hypothesis of any kind, not even in a "basic stage". Only the deadline for June 16 exists right now. I'm happy he did not reject me working on it. He encourages me to work harder. I got a "Great job so far!" compliment. === Google's "AI Mode" === This part was 100% AI-generated: {{quote|What you are describing is a core concept in the philosophy of science called Falsification. It was popularized by Karl Popper, who argued that science doesn't progress by "proving things right," but by rigorously trying to prove things wrong and failing to do so.}} 2xjzm1kcpkgmdmqvxjjlwiqwwrg4x5q 2807086 2806987 2026-04-30T07:43:57Z ThinkingScience 3061446 /* April 29, 2026 */ == April 30, 2026 == posting this text is self-therapy? 2807086 wikitext text/x-wiki Template Links: * {{tl|Draft}} * {{tl|underconstruction}} '''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].''' == "Diary" == == April 18, 2026 == A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a "diary" that will be only regarding the project. Turned "me language" into expressing that everyone is welcome, that I don't "own" [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says "I did this" "I did that" should be gone. I think this was an improvement of some sort. Plan for next edits on the draft page: Add an <nowiki>" == Old Methodology needing updating == "</nowiki> where I will add old methodologies where I had not planned ahead too good and the "Do no harm" I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago. Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future? == April 20, 2026 == Why does it seem like I'm the only one using the word "methodology"? Did the [[Wikiversity:Research ethics]] mention it? * I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now. == April 21, 2026 == I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the "Do no harm". Considering I have almost not developed anything but I still gotta work on this...to do... == April 22, 2026 == I am yearning / looking forward to working on video notes in a "Do no harm" way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different. == April 27, 2026 == I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more. Today I woke up being inspired by that: * Is there a hypothesis or a number of hypotheses related to the idea? ** Can this idea be proven false? Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity. Deadlines etc. can help me keep moving... What a hypothesis is: "I predict this will happen" and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect. [[Operationalization]] is then also needed to make "ambiguous ideas" measurable. My father remembers that I "wanted to save the world" but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea... == April 29, 2026 == Just made my father aware of this "idea" that so far doesn't have a published hypothesis of any kind, not even in a "basic stage". Only the deadline for June 16 exists right now. I'm happy he did not reject me working on it. He encourages me to work harder. I got a "Great job so far!" compliment. === Google's "AI Mode" === This part was 100% AI-generated: {{quote|What you are describing is a core concept in the philosophy of science called Falsification. It was popularized by Karl Popper, who argued that science doesn't progress by "proving things right," but by rigorously trying to prove things wrong and failing to do so.}} == April 30, 2026 == Was gonna start writing on the Colloquium again and composed a large message to reply/'talk to' Jtneill but I got input from "AI Mode" and then I realized after a while, maybe what I'm trying to ask I can find out with the help of "AI Mode" leading me to the right resources? ie. formatting on talk pages...how important is that? Prioritizing... Also replying to everything? Is it really needed? Would I like to be known as "Needy ThinkingScience"? I can't do anything on my own? So yeah, I started thinking maybe I can focus on doing more. Starting by talking less and the tools we have available can fix a lot anyway, LLMs are great but designed in interesting and challenging ways! Do I even need a mentor or do I '''just think I need one'''? I am very much a needy guy in terms of hand-holding and being related to a social context "historically" in my life, or at least that's what I thought. What I think I need for myself may be completely wrong. People in talk pages I interpreted as saying that I shouldn't put obstacles in my path by being mean to myself and putting pressure on myself to do certain things in a certain amount of time but I guess I couldn't stop myself! I also need a social context but would I just drag others into my "bad routines"? It all depends on who one interacts with...which now makes me think about "the idea" again. I guess I should see this as a warning sign :) My father suggested peer-review related activities and here I am looking inward and isolating myself. I don't know... ek7689tn79ooa0si4cni5wrtuuvnrpv 2807089 2807086 2026-04-30T07:50:34Z ThinkingScience 3061446 /* April 30, 2026 */ let actions speak a bit and any mistakes comment on my user talk page 2807089 wikitext text/x-wiki Template Links: * {{tl|Draft}} * {{tl|underconstruction}} '''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].''' == "Diary" == == April 18, 2026 == A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a "diary" that will be only regarding the project. Turned "me language" into expressing that everyone is welcome, that I don't "own" [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says "I did this" "I did that" should be gone. I think this was an improvement of some sort. Plan for next edits on the draft page: Add an <nowiki>" == Old Methodology needing updating == "</nowiki> where I will add old methodologies where I had not planned ahead too good and the "Do no harm" I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago. Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future? == April 20, 2026 == Why does it seem like I'm the only one using the word "methodology"? Did the [[Wikiversity:Research ethics]] mention it? * I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now. == April 21, 2026 == I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the "Do no harm". Considering I have almost not developed anything but I still gotta work on this...to do... == April 22, 2026 == I am yearning / looking forward to working on video notes in a "Do no harm" way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different. == April 27, 2026 == I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more. Today I woke up being inspired by that: * Is there a hypothesis or a number of hypotheses related to the idea? ** Can this idea be proven false? Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity. Deadlines etc. can help me keep moving... What a hypothesis is: "I predict this will happen" and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect. [[Operationalization]] is then also needed to make "ambiguous ideas" measurable. My father remembers that I "wanted to save the world" but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea... == April 29, 2026 == Just made my father aware of this "idea" that so far doesn't have a published hypothesis of any kind, not even in a "basic stage". Only the deadline for June 16 exists right now. I'm happy he did not reject me working on it. He encourages me to work harder. I got a "Great job so far!" compliment. === Google's "AI Mode" === This part was 100% AI-generated: {{quote|What you are describing is a core concept in the philosophy of science called Falsification. It was popularized by Karl Popper, who argued that science doesn't progress by "proving things right," but by rigorously trying to prove things wrong and failing to do so.}} == April 30, 2026 == Was gonna start writing on the Colloquium again and composed a large message to reply/'talk to' Jtneill but I got input from "AI Mode" and then I realized after a while, maybe what I'm trying to ask I can find out with the help of "AI Mode" leading me to the right resources? ie. formatting on talk pages...how important is that? Prioritizing... Also replying to everything? Is it really needed? Would I like to be known as "Needy ThinkingScience"? I can't do anything on my own? So yeah, I started thinking maybe I can focus on doing more. Starting by talking less and the tools we have available can fix a lot anyway, LLMs are great but designed in interesting and challenging ways! Do I even need a mentor or do I '''just think I need one'''? I am very much a needy guy in terms of hand-holding and being related to a social context "historically" in my life, or at least that's what I thought. What I think I need for myself may be completely wrong. People in talk pages I interpreted as saying that I shouldn't put obstacles in my path by being mean to myself and putting pressure on myself to do certain things in a certain amount of time but I guess I couldn't stop myself! I also need a social context but would I just drag others into my "bad routines"? It all depends on who one interacts with...which now makes me think about "the idea" again. I guess I should see this as a warning sign :) My father suggested peer-review related activities and here I am looking inward and isolating myself. I don't know... :I figured today I begin with {{quote|April 20th experiment, "AI Decisions, sure. AI-generation NEVER"}} 4cd0mzyclyg5goh9eqadt93j7qmru1f 3 329203 2807026 2805654 2026-04-29T16:51:03Z Atcovi 276019 +sec 2807026 wikitext text/x-wiki ==What We've Got (4/29/2026)== 15 posters collected, strong enough to build the poster. AI-generated overview: 0rmmm5l1wtelibeeo3x2xkp2nlwbvqr 2807027 2807026 2026-04-29T16:51:42Z Atcovi 276019 /* What We've Got (4/29/2026) */ 2807027 wikitext text/x-wiki ==What We've Got (4/29/2026)== 15 posters collected, strong enough to build the poster. AI-generated overview: 🔥 🧩 What you’ve actually built (this is the key insight) Your paper now has a clear, high-level model: '''🧠 Core Argument (clean version)''' OGM is a trait-like cognitive vulnerability * → that is amplified under stress and depression * → impairs problem solving + increases rumination + reduces future thinking * → leads to hopelessness * → contributes to suicidal ideation '''⚠️ With critical nuance (THIS is what makes it strong)''' From your notes: * OGM ≠ universal predictor (Crane 2016) * OGM requires stress activation (Stange 2013) * OGM strongest in clinical populations (Zhu, Kaviani, Arie) * OGM persists beyond depression (Hallford, Weiss-Cowie) 👉 That combination = real theoretical sophistication '''📊 🔍 Quality Assessment''' 🔴 Core SI Evidence → STRONG You have: * Zhu (independent predictor) * Kaviani (mechanism → SI) * Arie (behavior) * Jiang (integrated model) 👉 This alone is enough for a poster. 🟡 Mechanisms → VERY STRONG * Problem solving * Hopelessness * Rumination * CaR-FA-X 👉 This is where your review becomes high-level 🔵 Theory / Vulnerability → COMPLETE * Trait vs state ✔ * Remission ✔ * High-risk ✔ * ⚫ Contradictions → EXCELLENT You explicitly included: * Null findings * Trauma critique * Population differences 6i1yxilg9jjuxkg97bl5m7ncup8eg5m 0 329322 2806996 2806913 2026-04-29T14:24:02Z Megumi Fazakerley 3069053 save 5 2806996 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''Figure 1: Papyrus 47 folio 7''' [[File:P47 folio 7 – Rev 13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''Figure 2: P46 folio 62'''[[File:P47 folio 7 – Rev 13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. '''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)'''[[File:Visual Parody.png|alt=Figure 3: Side-by-Side Comparison of ΧΞϚ (in P47 f.7) and ΧΡϹ (in P46 f.62)|frameless|800x800px]]Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.<ref>The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.</ref> '''Figure 4: Letter – Object Visual Association in English''' [[File:Sammy Snake in Classroom.jpg|frameless|449x449px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''Figure 5: Hebrew Letter א''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|last=Wikipedia contributors|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''Figure 7: Examples of Common Modern Signs'''[[File:Modern Common Sings.png|frameless|800x800px]]Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. The image of the Letterland character, Sammy Snake, is used by permission of Letterland, Riverbridge House, Guildford Road, Leatherhead, Surrey, KT22 9AD, UK. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} 2v57lo5kdq6onk53yvqyut2zhpe46c3 2806997 2806996 2026-04-29T14:32:36Z Megumi Fazakerley 3069053 Save 6 2806997 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''Figure 1: Papyrus 47 folio 7''' [[File:P47 folio 7 – Rev 13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''Figure 2: P46 folio 62'''[[File:P47 folio 7 – Rev 13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. '''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)'''[[File:Visual Parody.png|alt=Figure 3: Side-by-Side Comparison of ΧΞϚ (in P47 f.7) and ΧΡϹ (in P46 f.62)|frameless|800x800px]]Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.<ref>The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.</ref> '''Figure 4: Letter – Object Visual Association in English''' [[File:Sammy Snake in Classroom.jpg|frameless|449x449px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''Figure 5: Hebrew Letter א''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''Figure 7: Examples of Common Modern Signs'''[[File:Modern Common Sings.png|frameless|800x800px]]Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. The image of the Letterland character, Sammy Snake, is used by permission of Letterland, Riverbridge House, Guildford Road, Leatherhead, Surrey, KT22 9AD, UK. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} 8mfdrmlybz03myedwwspoiipsdatkbn 2806998 2806997 2026-04-29T14:48:06Z Megumi Fazakerley 3069053 Save 7 2806998 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''Figure 1: Papyrus 47 folio 7''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 1. P47 folio 7|left|frameless|1247x1247px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''Figure 2: P46 folio 62''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 2: P46 folio 62|left|frameless|1247x1247px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. '''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)''' [[File:Visual Parody.png|left|frameless|800x800px]] Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.<ref>The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.</ref> '''Figure 4: Letter – Object Visual Association in English''' [[File:Sammy Snake in Classroom.jpg|frameless|449x449px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''Figure 5: Hebrew Letter א''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''Figure 7: Examples of Common Modern Signs''' [[File:Modern_Common_Sings.png|left|frameless|800x800px]] Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. The image of the Letterland character, Sammy Snake, is used by permission of Letterland, Riverbridge House, Guildford Road, Leatherhead, Surrey, KT22 9AD, UK. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} 9gsmlekzrtduh3ohm64uahn8umbayhy 2806999 2806998 2026-04-29T14:56:46Z Megumi Fazakerley 3069053 Save 8 2806999 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''Figure 1: Papyrus 47 folio 7'''[[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1231x1231px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''Figure 2: P46 folio 62''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1231x1231px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. '''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)'''[[File:Visual_Parody.png|frameless|790x790px]] Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.<ref>The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.</ref> '''Figure 4: Letter – Object Visual Association in English''' [[File:Sammy Snake in Classroom.jpg|frameless|449x449px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''Figure 5: Hebrew Letter א''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless|418x418px]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''Figure 7: Examples of Common Modern Signs'''[[File:Modern_Common_Sings.png|frameless|790x790px]] Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. The image of the Letterland character, Sammy Snake, is used by permission of Letterland, Riverbridge House, Guildford Road, Leatherhead, Surrey, KT22 9AD, UK. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} kqzstwj8eqrjr8jb4lj3m3oxv0r1smm 2807000 2806999 2026-04-29T15:07:28Z Megumi Fazakerley 3069053 Save 9 2807000 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''<small>Figure 1: Papyrus 47 folio 7</small>''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''<small>Figure 2: P46 folio 62</small>''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. <small>'''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)'''</small> [[File:Visual_Parody.png|frameless|800x800px]] Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.<ref>The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.</ref> '''<small>Figure 4: Letter – Object Visual Association in English</small>''' [[File:Sammy Snake in Classroom.jpg|frameless|449x449px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''<small>Figure 5: Hebrew Letter א</small>''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless|418x418px]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''<small>Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain</small>''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''<small>Figure 7: Examples of Common Modern Signs</small>''' [[File:Modern_Common_Sings.png|frameless|800x800px]] Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. The image of the Letterland character, Sammy Snake, is used by permission of Letterland, Riverbridge House, Guildford Road, Leatherhead, Surrey, KT22 9AD, UK. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} hq3mt8pun24tjcc0sho44r1h650pv98 2807091 2807000 2026-04-30T08:35:42Z Megumi Fazakerley 3069053 Save 10 (replaced a copyrighted image with CC0 image) 2807091 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''<small>Figure 1: Papyrus 47 folio 7</small>''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''<small>Figure 2: P46 folio 62</small>''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. <small>'''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)'''</small> [[File:Visual_Parody.png|frameless|800x800px]] Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake'. Its copyrighted image cannot be included here, but the picture (Figure 4) below illustrates how it works in classroom. '''<small>Figure 4: Letter – Object Visual Association in English</small>''' [[File:Silly Snake in Class.jpg|frameless|600x600px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''<small>Figure 5: Hebrew Letter א</small>''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless|418x418px]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''<small>Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain</small>''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''<small>Figure 7: Examples of Common Modern Signs</small>''' [[File:Modern_Common_Sings.png|frameless|800x800px]] Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} c1o0ca6scc2byuoolyx2ct0lczqbb9b 2807092 2807091 2026-04-30T08:51:27Z Megumi Fazakerley 3069053 Save 11 2807092 wikitext text/x-wiki {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Fazakerley | orcid1 = 0009-0009-4470-1435 | first1 = Megumi | affiliation1 = SIM | correspondence1 = megumi.fazakerley@sim.org | keywords = 666, pictogram, symbol, visual, interpretation, antichrist | license = CC-BY | abstract = This article proposes a visual-symbolic interpretation of the number χξϛ (666) in Revelation 13:18 as a pictogram. While most scholarly attention continues to focus on numerical symbolism, this paper suggests that the visual appearance of the Greek alphabetic numerals ΧΞϚ likely served as an additional layer of symbolic meaning to John and his original readers in first-century Asia Minor. Drawing on archaeological and cultural evidence, including the use of pictograms in first-century Ephesus, this study argues that pictographic perception formed part of the interpretative tool kit for the audience and that the distinctive zig-zag shape of Ξ in handwritten form plausibly evoked in their minds a symbolic association with a serpent, supporting the sustained narrative theme of deception and Satanic parody in Revelation and beyond. }} ==1. Introduction== 'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:<blockquote>You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.<ref>{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}</ref></blockquote>This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider? The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (''thērion'', beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.<ref>{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}</ref><ref>{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 February 2011|website=The Gospel Coalition}}</ref> Another proposal has been made to interpret the number ''visually'' by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.<ref>{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre & Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}</ref> Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw. == 2. Examining the Text in its Literary Context == The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it. === 2.1 Visual Nature of Revelation and of the Mark === Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of ''seen'' visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. In particular, χξϛ was the ''mark'' of the beast, i.e. a ''visual'' symbol of allegiance ''to be seen'' on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ ''looked'', i.e. its visual appearance to human eyes. === 2.2 Visual Form of ΧΞϚ and its Function as Pictogram === The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.<ref>{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}</ref> However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.<ref>{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}</ref> ==== 2.2.1 Greek Numeral System ==== Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&oldid=1296826253|title=Arabic numerals|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&oldid=1264812857|title=History of the Hindu–Arabic numeral system|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}</ref> The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&oldid=1222860822|title=Alphabetic numeral system|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}</ref> This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&oldid=1295786959|title=Greek numerals|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}</ref> [[File:666 in Greek Shorthand.png|frameless|42x42px]] is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below: {| class="wikitable" |+Table 1: Numerical Values of ΧΞϚ !Letter !Letter Name !Numerical Value |- |Χ |chi |600 |- |Ξ |xi |60 |- |Ϛ |stigma |6 |} Yet, it is their ''visual forms'' that need our particular attention, because the number was a ''visual'' symbol ''to be seen'' on the openly visible parts of the body of the beast-followers. ==== 2.2.2 Handwritten Form in Majuscule ==== At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. ==== 2.2.3 Σ (sigma) and Ϛ (stigma) ==== Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&oldid=1298495170|title=Sigma|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}</ref> As such, it is commonly found in many early manuscripts of the New Testament. The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> '''<small>Figure 1: Papyrus 47 folio 7</small>''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]] The text on the ninth line says: ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ) which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:<ref>{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 October 2022|archive-url=|website=Center for New Testament Restoration}}</ref> [[File:Koine_Majuscule_text.png|frameless|380x380px]] What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ). ==== 2.2.4 Nomina Sacra ==== Early Christians considered certain names and titles, like Θεός (''Theos'', God), Κύριος (''Kyrios'', Lord), Ἰησοῦς (''Iēsous'', Jesus), Χριστός (''Christos'', Christ), Υἱός (''Huios'', Son, referring to Jesus) and Πνεῦμα (''Pneuma'', Spirit, referring to the Holy Spirit), as ''nomina sacra'' (sacred names), to be treated with respect.<ref name=":0">{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&oldid=1270217331|title=Nomina sacra|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}</ref> Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals. Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&oldid=1283806604|title=Scribal abbreviation|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}</ref><ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&oldid=1254008920|title=Memorabilia (Xenophon)|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}</ref> In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that '''nomina sacra'' are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'<ref name=":0" /> It is, then, reasonable to estimate that origin of ''nomina sacra'' was early in the first century. That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of ''nomina sacra''. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (''Christos'', Christ). Initially, this practice was limited to only a handful of words, which are called ''nomina divina'' (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.<ref>{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}</ref> {| class="wikitable" |+Table 2: Nomina Divina ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Θεός |''Theos'' |God |ΘΣ |ΘΥ |- |Κύριος |''Kyrios'' |Lord |ΚΣ |ΚΥ |- |Ἰησοῦς |''Iēsous'' |Jesus |ΙΣ or ΙΗΣ |ΙΥ |- |Χριστός |''Christos'' |Christ |ΧΣ or ΧΡΣ |ΧΥ |- |Πνεῦμα |''Pneuma'' |Spirit referring to the Holy Spirit |ΠΝΑ |ΠΝΣ |} {| class="wikitable" |+Table 3: Later Additions to Nomina Sacra ! rowspan="2" |Greek ! rowspan="2" |Transliteration ! rowspan="2" |English ! colspan="2" |Abbreviation |- !(Nominative Case) !(Genitive Case) |- |Πατήρ |''Patēr'' |Father |ΠΗΡ |ΠΡΣ |- |Σωτήρ |''Sōtēr'' |Saviour |ΣΗΡ |ΣΡΣ |- |Σταυρός |''Stauros'' |Cross |ΣΤΣ |ΣΤΥ |- |Μήτηρ |''Mētēr'' |Mother referring to Mary |ΜΤΡ |ΜΡΣ |- |Ἰσραήλ |''Israēl'' |Israel |ΙΗΛ | |- |Ἄνθρωπος |''Anthrōpos'' |Man in the phrase 'Son of Man' |ΑΝΟΣ |ΑΝΟΥ |- |Ἰερουσαλήμ |''Ierousalēm'' |Jerusalem |ΙΛΗΜ | |- |Οὐρανός |''Ouranos'' |Heaven |ΟΥΝΟΣ |ΟΥΝΥ |} Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.<ref>{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}</ref> As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically. '''<small>Figure 2: P46 folio 62</small>''' [[File:P47_folio_7_–_Rev_13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]] In the manuscript, the text on the ninth line reads: ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν) which looks more like: [[File:Koine Majuscule text 2.png|frameless|380x380px]] where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as [[File:God's Son Jesus Christ.png|frameless|122x122px]]. The point to note here is that if the first-century readers were accustomed to seeing [[File:Christ in shorthand 2.png|frameless|22x22px]] or [[File:Christ in abbreviated form.png|frameless|33x33px]], written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark. ==== 2.2.5 ΧΞϚ as Visual Parody ==== The mark of the beast in Revelation 13:18 was written as [[File:666 in Greek Shorthand.png|frameless|35x35px]]. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between [[File:666 in Greek Shorthand.png|frameless|35x35px]] and [[File:Christ in abbreviated form.png|frameless|33x33px]] could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts. <small>'''Figure 3: Side-by-Side Comparison of''' [[File:666 in Greek Shorthand.png|frameless|35x35px]] '''(in P47 f.7) and''' [[File:Christ in abbreviated form.png|frameless|35x35px]] '''(in P46 f.62)'''</small> [[File:Visual_Parody.png|frameless|800x800px]] Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.<ref>Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see <nowiki>https://www.letterland.com/company</nowiki>.</ref> In their system, the letter S is taught as 'Sammy Snake'. Its copyrighted image cannot be included here, but the picture (Figure 4) below illustrates how it works in classroom. '''<small>Figure 4: Letter – Object Visual Association in English</small>''' [[File:Silly Snake in Class.jpg|frameless|600x600px]] There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed. If, then, people in the ancient Greek speaking world considered [[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]] serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', [[File:666 in Greek Shorthand.png|frameless|40x40px]] ([[File:Parody-christ in shorthand.png|frameless|42x42px]]) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of [[File:Christ in shorthand.png|frameless|40x40px]] ([[File:Christ in shorthand Koine handwriting.png|frameless|33x33px]]), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|16x16px]]) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. Such a parody by Satan, in mockery to God and for deception of people, fits the immediate context of the vision, where the beast mimics divine power (Rev. 13:3–4). This theme of Satanic deception by mimicry is not confined to chapter 13, but it is sustained throughout Revelation (Rev. 2:2; 3:9; 16:13–14; 17:3 vs 12:1; 17:8; 19:20; 20:7–8), and beyond, going back to the Gospels (e.g. Matt. 7:15; 24:24; Luke 6:26; John 10:11–13) and the rest of the New Testament (e.g. 2 Cor. 11:12–15; 1 Tim. 4:1; 1 John 4:1), and even further to the Old Testament (e.g. Gen. 3:1; Deut. 13:1–5, Micah 3:5). Not too long ago, John had written to his readers in his care about 'antichrists' who had already arrived in 'the last hour' (1 Jn. 2:18). It should be noted that in Greek the sense of ἀντίχριστος is more an impostor than an opponent of Χριστός, as the core sense of the preposition ἀντί is 'opposite', in the sense that the right hand is opposite the left hand and two people facing each other are opposite each other. Neither hostility nor conflict are implied or necessary for the relationship between the two opposite each other. The preposition simply denotes 'various types of correspondence ranging from replacement to equivalence'.<ref>{{Cite book|title=A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 4th ed.|last=Bauer|first=Walter, Frederick William Danker, William Frederick Arndt and Felix Wilbur Gingrich|publisher=University of Chicago Press|year=2021|location=Chicago|pages=76}}</ref> Its sense, therefore, is not so much 'against' (opposition) as 'instead of' or 'in place of' (replacement). Thus, an antichrist is someone who takes the Christ's place and acts like him. The testimony of the Bible is that Jesus is the Christ, and no-one else. Thus, someone else taking his place and acting like him is a hypocrite, a pretender and an unworthy fake. The idea of a serpent-like inner essence with a superficial outer resemblance to Christ would have been an effective metaphor of antichrist to John and his readers, reminding them of Jesus' earlier metaphors like 'ferocious wolves in sheep's clothing' or 'the abomination that causes desolation standing in the holy place' where it does not belong, when he warned them about the false teachers and many who would come in his name, claiming to be him (Matt. 7:15; 24:4,15). == 3. Considering Evidence from the Historical Context == Revelation is set in the historical milieu of first-century Asia Minor. John was on the island of Patmos 'in the suffering and … patient endurance' 'because of the word of God and the testimony of Jesus' (1:9). From Irenaeus (c. AD 130–202, a disciple of Polycarp, who himself was a disciple of John) to Eusebius (c. AD 260–340) and Jerome (c. AD 347–420), early testimonies consistently place John in a prominent role of Christian leadership based in Ephesus in the latter part of the first century. The broad consensus in biblical scholarship today is that John was banished to Patmos under persecution during the reign of Caesar Domitian (AD 81–96).<ref>{{Cite book|title='Introduction: the circumstances of the book', in The Message of Revelation (electronic edition for Olive Tree Bible software)|last=Wilcock|first=Michael|publisher=InterVarsity Press|year=1991|location=Downers Grove, IL|pages=}}</ref> === 3.1 Cultural Diversity and Pictographic Influence === Asia Minor in the first century belonged to the Graeco-Roman world, which was unified by the shared heritage of the Hellenistic culture including the Greek language and the political and military rule by the Roman Empire. Apart from those common factors, however, the area was also characterised by diversity as a melting pot of peoples, cultures and religious traditions, with influences from Anatolian, Greek, Roman, Egyptian, Babylonian, Persian and of course Jewish traditions. Among its inhabitants, a good number would have had cultural backgrounds familiar with pictographic literacy, in which written symbols represent an object or an idea. This was in contrast with all European languages, which used phonetic scripts, where each letter represents a sound. Egyptian hieroglyphics and Mesopotamian cuneiforms are the best-known examples of pictographic writing system that widely influenced the ancient world. While itself phonographic, even the Hebrew script, considered by some to be the oldest of all alphabets,<ref>{{Cite web|url=https://www.sciencenews.org/article/oldest-alphabet-identified-hebrew|title=Oldest alphabet identified as Hebrew|last=Bower|first=Bruce|date=19 November 2016|website=Science News}}</ref> had its origin in hieroglyphic pictography.<ref>{{Cite web|url=https://bible.ca/manuscripts/English-Hebrew-chart-worlds-oldest-alphabet-Douglas-Petrovich-original-first-Proto-Consonantal-Sinaitic-Canaanite-Script-Pictograms-Photograms-Echograms-Egyptian-Hieroglyphics-Avaris-Tel-el-Daba-1859-1842BC.jpg|title=Biblical 'Hebrew to English' Alphabet — Hebrew is the first and oldest alphabet: 1859 BC|last=Rudd|first=Steve|website=The Interactive Bible|access-date=19 May 2025}}</ref> It is, then, not too difficult to imagine Jewish parents teaching their young children Hebrew letters by encouraging them to pay attention to the letter's shape ''visually'' and to picture in their mind what it looks like, for example, by saying, 'The first letter א looks like the head of אֶלֶף (elep̱; ox, cow, cattle) with two horns sticking up.' If this imagination has any merit, it is more than likely that most Hebrew readers had at least some experience in ''visually'' associating letters to images of objects. '''<small>Figure 5: Hebrew Letter א</small>''' [[File:Pictographic Origin of Hebrew Letter א (aleph).png|frameless|418x418px]] It is, then, plausible that upon encountering ΧΞϚ as a 'mark' (i.e. a visual symbol), John and a good number of his original readers would have been open to interpreting it, not only phonetically and numerically, but also pictographically. === 3.2 Visual Symbolism in Mysticism and Magic at Ephesus === First-century Asia Minor was also characterised by the wide-spread influence of magic and mysticism, and the use of visual symbols in magical or mystical circles in ancient Hellenistic world is widely attested. For example:<blockquote>A large number of magical signs and symbols appear on amulets, gems, and tablets.... In Gnosticism they were also taken over by Christian magic (Book of Jeu, Pistis Sophia).<ref>{{Cite web|url=https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/magic-magic-greco-roman-antiquity|title=Magic: Magic in Greco-Roman Antiquity|website=Encyclopedia of Religion, Encyclopedia.com|access-date=19 May 2025}}</ref></blockquote>Indeed, Acts 19 shows the pervasive nature of magical activity in first-century Ephesus. As many responded to the gospel and turned to Christ, they came forward to burn their magical books. Obviously, none of those texts from Ephesus survived. However:<blockquote>A magician's kit, probably dating from the third century, was discovered in the remains of the ancient city of Pergamon in Anatolia.… The find consisted of a bronze table and base covered with ''symbols'', a dish (also decorated with symbols), a large bronze nail with ''letters inscribed'' on its flat sides, two bronze rings, and three black polished stones ''inscribed with the names'' of supernatural powers.<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Magic_in_the_Greco-Roman_world&oldid=1286605948|title=Magic in the Greco-Roman world|date=20 April 2025|website=Wikipedia, The Free Encyclopedia|postscript=, (emphases mine to indicate use of symbols and markings)}}</ref></blockquote>A picture painted by the archaeological evidence, then, is that John and his readers would have been familiar with people who used various types of visual markings for magical-mystical purposes, i.e. for carrying encoded meanings of spiritual, rather than historical or factual, nature. When John saw in the vision those people who had the mark of the beast on their right hands or foreheads, he would have been quick to identify them as belonging to such circles, who were inclined to look for a ''spiritual'' meaning of their symbol, such as the name, nature or characteristic of the being they worship, at least just as much as to try linking it to a contemporary historical figure. === 3.3 Pictogram from First-Century Ephesus and Visual Symbolism in First-Century Asia Minor === A particular archaeological artefact from first-century Ephesus in Asia Minor is discussed in a paper published in an orthopaedic journal in 2013. They write:<blockquote>The traditional approach to history based on accentuating the most outstanding political, military and cultural events is increasingly opposed by a more complete vision of the past through a sociological approach inspired by the fate of ordinary people and their daily lives. An ordinary everyday experience was recorded on this advertising sign engraved in the marble of the ancient Ephesus.<ref name=":1">{{Cite web|url=https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3764288|title=The pictogram of the pes planus from the first century AD|last=Wokaunn|first=Mario, Stella Fatović-Ferenčić and Michele Mikolaučić|website=International orthopaedics|series=vol. 37.9 (2013)|pages=1871–1873|doi=10.1007/s00264-013-2020-4}}</ref></blockquote>It is an image of a foot in an advertising sign. Its purpose is to persuade people to walk to the advertised establishment and to show a direction to its location. The authors are right to identify it as a pictogram, as it was a visual symbol that conveys a particular message. For them, however, their main interest lies with the image itself rather than its sociolinguistic function as a pictogram. The image is such a realistic illustration that they believe it to be an imprint (or perhaps a very good drawing). And, by applying modern diagnostic methodology, they confirmed it as a description of a flat foot. They conclude that this pictogram is uniquely valuable as 'the oldest known illustration of this particular pathology,'<ref name=":1" /> as historical records of flat foot is extremely rare, despite flat foot being common today and present in all ethnic groups and in all time periods.<ref>It would make sense, if flat feet were not considered worth discussing, that little historical evidence remain to document it. If people thought the condition was too mundane, obvious or otherwise pointless to talk about, they would not have written about it.</ref> It is interesting to note that a condition as ubiquitous as flat feet could have escaped documentation universally for so long. Perhaps it was considered so ordinary, obvious and unremarkable, people did not see any value for documenting it for themselves or for posterity. It is a helpful reminder that the lack of documented evidence does not mean non-existence. However, for the purpose of this present consideration, it is not the image itself or what the image describes that matters. What matters is that this image functioned socio-linguistically as a message-carrying symbol, i.e. a pictogram, in first-century Ephesus. This particular pictogram was engraved into the Marble Road of Ephesus and survived as a part of an advertisement. It offers a valuable insight into the life of ordinary people there. As Ephesus was one of the seven cities addressed in Revelation, this insight leads to the conclusion that the people in first-century Asia Minor were pictographically literate. Though they used Greek with its phonetic alphabet for writing, they were also accustomed to the practice of interpreting written symbols for their visual associations to objects or ideas. John and his readers, therefore, would have been as ready to interpret the mark of the beast as a pictogram as to interpret it as a number puzzle. === 3.4 Literacy and Prevalence of Symbol Usage by Early Christians in Graeco-Roman World === Christians used symbols from early days. Perhaps the best-known example is the symbol of fish, as shown in Figure 6 below.<ref>{{Cite web|url=https://www.researchgate.net/figure/Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis_fig3_366561316|title=Fish symbol and maritime motifs on late antique lamps from Central Balkans|date=6 November 2022|website=ResearchGate|page=275|doi=10.5937/zrffp52-41296|access-date=28 May 2025}}</ref> '''<small>Figure 6: Funerary stele of Licinia Amias, early 3rd century AD from the area of the Vatican necropolis, Rome, National Archaeological Museum, inv. no 67646 © public domain</small>''' [[File:Funerary-stele-of-Licinia-Amias-early-3-rd-AD-From-the-area-of-the-Vatican-necropolis.png|frameless|500x500px]] It was used as a mark of Christian identity, with ἸΧΘΥΣ (or ἸΧΘΥϹ with a lunate sigma) being an acronym for Ἰησοῦς Χρῑστός Θεοῦ Υἱός Σωτήρ, which translates into English as 'Jesus Christ, Son of God, Saviour.'<ref>{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Ichthys&oldid=1288777270|title=Ichthys|date=4 May 2025|website=Wikipedia, The Free Encyclopedia}}</ref> This and other similar symbols were all pictograms, i.e. images that carried their encoded messages. The literacy rate in the first-century Hellenistic world under the Roman rule is variously estimated. While some suggest less than 15%,<ref>{{Cite web|url=https://www.academia.edu/53973811/Ancient_literacy|title=Ancient Literacy|last=Mitch|first=David|website=Economics of Education Review|series=14.1 (1995)|page=96|doi=10.1016/0272-7757(95)90111-6|access-date=28 May 2025}}</ref> others propose about 30%,<ref>{{Cite web|url=https://bmcr.brynmawr.edu/1992/1992.03.07/|title=Literacy in the Roman World|last=Williamson|first=Callie|website=Bryn Mawr Classical Review|series=1992.03.07|access-date=28 May 2025}}</ref> and still others claim as high as 80%.<ref>{{Cite web|url=https://www.learnancientrome.com/what-was-the-literacy-rate-in-ancient-rome/|title=What Was The Literacy Rate In Ancient Rome|last=Rideout|first=Moshe|date=1 November 2023}}</ref> Probably, it is safe to think more broadly that a larger proportion of people were only semi-literate at best, if not completely illiterate. In such a context, the advantage of symbolic images was that they could convey their message to everyone, even to those who are not fully competent in reading and writing. In UK today, there are people who are less than fully literate or competent in numeracy (e.g. younger children), but when they see in town a sign that says (usually in red) '999', they like everyone else will likely know what it means, because they will recognise it, not as a number, but as an image symbolising emergency service. Likewise, when they see a sign that says 'EXIT' or 'TOILET' (often accompanied by a simple illustration as shown in Figure 7 below), they are more likely to perceive the whole sign as a symbol, rather than as a word, and get the intended message. '''<small>Figure 7: Examples of Common Modern Signs</small>''' [[File:Modern_Common_Sings.png|frameless|800x800px]] Similarly, in today's world, not all shoppers may be accomplished arithmeticians, but they can still go to markets or supermarkets and make sense of the price tags on what they want to buy. Presumably, not everyone in the first-century Graeco-Roman world were fully competent in numeracy, but they would have been familiar enough with the appearance of numbers in common shorthand to be able to picture ΧΞϚ ('666') in their minds when they heard the number, even if they might have struggled to read or write it fully in words ἑξακόσιοι ἑξήκοντα ἕξ ('six hundred and sixty-six'). In the context of the first-century Graeco-Roman world, where many people were less than fully literate either in reading and writing or in numeracy, symbols would have been an effective means of communication. It would seem much easier to imagine people there being able to make sense of ΧΞϚ pictographically as a snake pretending to look like Christ than to imagine them capable of spelling out 'Nero Caesar', transliterating that by using Hebrew letters, converting each letter to a number according to the rules of Hebrew gematria, and finally adding up all the numbers to conclude that the number must refer to him. == 4. Conclusion == In the absence of documentary testimony evidencing ancient readers viewing ΧΞϚ ([[File:666 as it appears in many manuscripts.png|frameless|34x34px]]) as a pictogram, its pictographic interpretation as a Satanic parody of ΧΡϹ ([[File:Christ as the word appears in many manuscripts.png|frameless|36x36px]]) cannot be proven. However, it is not disproved, either. In fact, the literary evidence of the visual-symbolic nature of Revelation and the archaeological evidence that points to the pictographic literacy of John and his original readers provide support for it. The zig-zag snake-reminding shape of handwritten Ξ ([[File:Ξ in Handwritten Koine Manuscripts.png|frameless|14x14px]]), together with Χ and Ϛ sandwiching it in the middle, form a striking image that invites interpretation, not only as a number, but also as a symbol of Satanic deception. This proposal should encourage further exploration of the visual dimension of the text embedded in a document so richly symbolic as Revelation. 'Pictograms have transcended their ancient origins to become a universal language in modern graphic design.'<ref>{{Cite web|url=https://outrejournal.com/pictograms-history-evolution-graphic-design/|title=Pictograms in Graphic Design: A Universal Language|website=OutreJournal.com|access-date=13 May 2025}}</ref>An appeal of pictograms is that they are both universal and timeless. Taken pictographically, ΧΞϚ continues to speak across time and cultures. Satan sets himself against God, but he knows he is no match against God. So, he turns his attention to people, the crown of God's creation. He can employ a full-frontal attack approach aiming at our destruction (Rev. 12:17). More typically, however, his age-old strategy is through deception, aiming at persuading us to misplace our trust in him instead of God, as it happened with Adam and Eve in the garden. On the one hand, the numerical interpretation of ΧΞϚ sounds the alarm for the former by identifying a specific historical individual, like Nero Caesar, bent on conquest to force God's people to shift our allegiance away from God to him. On the other hand, the visual-symbolic interpretation can serve as an extra layer in the multi-layered caution, alerting us to the ongoing danger of the latter, that we might be vigilant. ==Additional information== ===Acknowledgements=== Scripture quotations taken from the Holy Bible, New International Version Anglicised Copyright © 1979, 1984, 2011 Biblica. Used by permission of Hodder & Stoughton Ltd, an Hachette UK company. All rights reserved. ‘NIV’ is a registered trademark of Biblica UK trademark number 1448790. I would like to thank Dr. Volker Glißmann for reading this article at different stages of writing and offering valuable advice and encouragement. ===Competing interests=== No competing interest. ==References== {{reflist|35em}} k9k9japg3dzqnzxh9x8s8j2n5le89y1 6 329347 2806986 2026-04-29T12:07:28Z Young1lim 21186 {{Information |Description=FF Timing (20260427 - 20260421) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2806986 wikitext text/x-wiki == Summary == {{Information |Description=FF Timing (20260427 - 20260421) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} ds3cnsxs64m78e7hm5rqx2cmmmaf5bp 6 329348 2806990 2026-04-29T13:16:49Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260429 - 20260428) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2806990 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260429 - 20260428) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} em9o07qp3k8g5amlzss0gj0f6o795ek 6 329349 2806992 2026-04-29T13:26:48Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260429 - 20260428) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2806992 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260429 - 20260428) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3ld8e3p5an4f6qvgj2tqwvukxqg9o33 6 329350 2806994 2026-04-29T13:42:48Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260429 - 20260428) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2806994 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260429 - 20260428) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 5vvrc7l0m1lb96v6ej5826upcblssq6 2 329351 2807029 2026-04-29T17:03:41Z Atcovi 276019 Create. 2807029 wikitext text/x-wiki ''Paper count:'' 15 (4/27/2026) == Research == === [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] === '''Background''' * Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner<ref>The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.</ref>. * Mentions number of studies associating OGM with suicide: "''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''" * [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓ * '''Research question(s)?''' No clear evidence to show how childhood trauma & OGM interact in the suicidal process and whether depression is a ''moderating'' effect. * '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals. * '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways. '''Method''' * '''356 Chinese participants'''. ** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. ** '''176 healthy individuals''' '''Measures''' * '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. * '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. * '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. * '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38. ** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. ** ↑ BSI-CV score ↑ SI * '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: "how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?". '''Results''' * Trauma → OGM, WSI, CSI * OGM → WSI + CSI * WSI → CSI * WSI → PSB * PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]''' WSI is strongly predictive of: * PSB * future SI intensity '''Discussion''' # '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically. # '''WSI''' = worst suicidal thoughts one has ever had [at a certain point]. # '''CSI''' = current point of suicidal ideation * [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together. * '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. ** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI. ** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. <-- appears to be relevant only with other paired vulnerabilities (ex, trauma). ** OGM as a stable trait of depression was more severe in the depression group vs. healthy control. ** Childhood trauma & OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model. ** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity. ** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation. ** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) & WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation. * Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy & MeST, both geared towards recalling memories in detail. * '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors. * '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior. === [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] === Crane et al. (2016) – Community adolescent cohort * Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16) * Tested whether OGM predicts depression/suicidality and moderates life stress * '''Findings:''' ** OGM did not predict depression, suicidal ideation, or self-harm ** OGM did not moderate the effect of life events ** Life events strongly predicted all outcomes * '''Interpretation:''' ** OGM may not function as a general vulnerability factor in community samples ** Likely only relevant in high-risk or depressed populations === Sumner et al. (2010) – Meta-analysis (OGM & depression) === * Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression * '''Findings:''' ** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up ** Effect remains '''even after controlling for baseline depression''' ** Overall effect size is '''small but significant (~1–2% variance explained)''' * '''Moderators:''' ** Stronger effect in '''clinically depressed samples vs nonclinical''' ** Stronger with '''shorter follow-up periods''' * '''Interpretation:''' ** OGM is a '''predictor of depression maintenance''', not just a correlate ** Likely acts as a '''vulnerability factor''', especially in high-risk groups {{Notice|OGM works under certain conditions (clinical / high-risk), not universally}} === Moore & Zoellner (2007) – Evaluative Review (OGM & trauma) === * Evaluative review of '''24 studies''' examining trauma exposure and OGM * '''Core question:''' Does trauma cause overgeneral memory? '''Findings''' * ❌ '''No consistent link''' between trauma exposure and OGM * ✅ OGM is more consistently linked to: ** '''Depression''' ** '''PTSD symptoms (intrusions, avoidance)''' * Trauma exposure alone is '''not sufficient''' to produce OGM * OGM appears more tied to '''psychopathology''', not the event itself '''Key nuance''' * Post-trauma '''symptoms''' (not the trauma itself) are what matter * Evidence across studies is '''mixed and methodologically inconsistent''' '''Interpretation''' * Challenges the classic “trauma → OGM” theory * Supports alternative view:<blockquote>OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)</blockquote> * Trauma alone isn’t enough — psychological response matters === Crane & Duggan (2009) – CSA onset & OGM in suicidal patients === * Clinical sample of '''49 patients with recurrent suicidal behavior''' * Examined whether '''age of onset of childhood sexual abuse (CSA)''' relates to OGM '''Findings''' * Earlier CSA onset → '''greater OGM''' (fewer specific, more categorical memories) * Effect remained after controlling for: ** depression ** verbal fluency * Presence vs absence of CSA '''did NOT differ in OGM''' ** (because sample already highly clinical) '''Interpretation''' * Not trauma itself → but '''timing of trauma (early development)''' matters * Supports idea that OGM develops when: ** memory systems are still forming ** avoidance becomes a learned coping style '''Insights''' # Analysis of OGM in a suicidal population. # Moore & Zoellner had trauma alone ≠ OGM, while Crane & Duggan added some trauma characteristics from early developmental trauma. # ''To take note'': Earlier trauma is associated with greater OGM '''within high-risk groups.''' === Champagne et al. (2016) – OGM in adolescent MDD === * Clinical adolescent sample (ages 11–18; current MDD, remitted MDD, never depressed) * Tested whether OGM is '''state (only during depression)''' or '''trait (persists after)''' '''Findings:''' * Adolescents with '''current AND remitted MDD''' showed more OGM than controls * No significant difference between current vs remitted groups * OGM present for both '''positive and negative cues''' '''Interpretation:''' * OGM is likely a '''trait-like vulnerability''', not just a temporary symptom * May contribute to '''risk of recurrence''' in depression '''1. Trait vs state question (very important)''' This paper directly answers:<blockquote>Is OGM just a symptom or a vulnerability?</blockquote>👉 Answer: '''likely vulnerability''' That’s big for your argument. ----'''2. Adolescent + clinical sample''' * Not just adults * Not just community 👉 Stronger relevance than many papers ----'''3. Fits your mechanism pathway''' Supports:<blockquote>OGM persists → increases long-term risk → can feed into suicidal ideation</blockquote>'''Limitations''' * Small sample (n=65 total) * Cross-sectional → can’t prove causality * Can’t tell if OGM came '''before''' depression or is a “scar” === Arie et al. (2008) – OGM, problem solving & suicidal behavior === * Clinical adolescent inpatient sample (suicidal vs nonsuicidal vs healthy controls) * Tested Williams’ model linking OGM, problem solving, and suicide '''Findings:''' * Suicidal adolescents showed: ** '''More OGM (less specific memory)''' ** '''Worse interpersonal problem-solving ability''' ** '''Higher hopelessness''' * OGM significantly correlated with: ** poorer problem solving ** higher hopelessness ** greater suicidal behavior * Negative childhood life events also linked to OGM and suicide '''Interpretation:''' * Supports pathway:<blockquote>OGM → poor problem solving → hopelessness → suicidal behavior</blockquote> * OGM may limit access to specific past experiences needed to solve problems effectively '''Insights''' * Study on population actually attempting suicide. * Points out a clear pathway * Suicidal group had lowest memory specificity, worst problem-solving abilities, and highest hopelessness '''Limitations''' * n=75, small sample * cross sectional, so causality can not be pinpointed * inpatient sample, so cannot be generalizable === Kaviani et al. (2011) – OGM, problem solving & suicidal ideation === * Clinical sample: depressed patients '''with vs without suicidal ideation''' * Tested links between OGM, problem solving, depression, and hopelessness '''Findings:''' * Suicidal ideation group: ** '''Less specific autobiographical memories (more OGM)''' ** '''More hopelessness''' ** '''Less effective problem-solving strategies''' * OGM associated with: ** poorer problem solving ** greater hopelessness * Evidence for a “cycle”:<blockquote>OGM → poor problem solving → hopelessness → suicidal ideation</blockquote> '''Interpretation:''' * OGM may contribute to suicidal ideation by '''impairing cognitive functioning''', especially problem solving * Supports a '''cognitive pathway to suicide risk''' '''Insights''' * Studies suicidal ideation directly. * Mechanism displayed: OGM --> problem solving --> hopelessness --> suicidal ideation * Within-depression comparison '''Limitations''' * n=40 * Cross-sectional, lacks causality * Specific country (Iran) === Zhu et al. (2025) – OGM & suicidal ideation (ML study) === * Clinical sample (n=88 depressed patients; with vs without suicidal ideation across severity levels) * Used '''AMT + machine learning + vocal features''' to distinguish suicidal ideation '''Findings:''' * Patients with suicidal ideation: ** '''More OGM (fewer specific memories)''' ** Especially impaired recall of '''positive memories''' * OGM strongly associated with '''severity of suicidal ideation''' * Crucially: ** '''OGM predicts suicidal ideation independently of depression severity''' * ML model showed: ** OGM features were key for identifying '''presence of suicidal ideation''' ** Depression severity features ≠ sufficient to predict SI '''Interpretation:''' * OGM is a '''specific cognitive marker of suicidal ideation''', not just a byproduct of depression * Supports:<blockquote>OGM contributes uniquely to suicidal ideation beyond general depression</blockquote>'''Insights''' * OGM is unique in contribution/specifically tied to SI. * Separated depression anxiety and suicidal ideation. OGM is not merely a correlate of depression, but a distinct cognitive vulnerability that contributes specifically to suicidal ideation. === Weiss-Cowie et al. (2023) — Meta-analysis on OGM & depression === '''🔑 Core findings (clean summary)''' * Massive meta-analysis ('''67 studies''') * Depressed individuals show: ** '''Less specific memories''' → ''(g = −0.73)'' ** '''More overgeneral memories''' → ''(g = 0.77)'' 👉 That’s '''medium–large effect sizes''' → very strong evidence ----'''🧠 Important conceptual findings''' '''1. OGM exists across:''' * Current depression * Subthreshold depression * Remitted depression 👉 Translation:<blockquote>OGM is not just a “currently depressed” phenomenon</blockquote> ----'''2. Trait vs state (VERY important for your work)''' * OGM persists even after remission * But is '''stronger during active depression''' 👉 This supports:<blockquote>OGM = '''trait vulnerability that gets amplified during depression'''</blockquote> ----'''3. Supports CaR-FA-X model''' * Capture & rumination * Functional avoidance * Executive dysfunction 👉 This is your '''mechanism backbone paper''' === Crane et. al (2016) === * Year: '''2016''' (so not “new,” but still solid) * Design: '''Large longitudinal cohort (ALSPAC)''' * Sample: ~5,700 adolescents * Timeline: ** OGM at age 13 ** Outcomes at age 16 '''Conclusions'''<blockquote>'''No meaningful relationship between OGM and later:'''</blockquote> * Depression * Suicidal ideation * Self-harm After controlling for confounds. This paper is basically saying:<blockquote>“OGM is NOT a general population risk factor for later psychopathology.”</blockquote>More precisely: * Works in: ** Clinical samples ** High-risk groups * '''Does NOT generalize to community samples''' '''Insights''' * high-quality longitudinal data '''1. Show inconsistency'''<blockquote>“While some studies show associations between OGM and suicidality, large community-based longitudinal evidence suggests no such relationship.”</blockquote> ----'''2. Introduce moderation / context argument'''<blockquote>“These discrepancies suggest that OGM may function as a vulnerability factor only under specific conditions (e.g., clinical populations, high-risk individuals).”</blockquote> ----'''3. Strengthen your theoretical argument''' This paper supports: * '''CaR-FA-X conditional vulnerability''' * '''Diathesis-stress framing''' === Hallford et. al (2022) === * Year: '''2022 → recent''' * Type: '''Systematic review + meta-analysis''' * Focus: OGM in '''remitted depression''' * Sample: '''17 studies''' '''Conclusion''' * People with '''remitted depression''': ** Recall '''fewer specific memories''' (g ≈ -0.31) ** Recall '''more overgeneral/categoric memories''' (g ≈ +0.25) 👉 These are '''small–moderate effects''', but consistent. ----<blockquote>OGM '''persists even after depression is gone'''</blockquote>This is HUGE. ----<blockquote>OGM may be a '''trait-like cognitive vulnerability''', not just a symptom</blockquote>The paper explicitly frames it as: * A '''risk factor for future depressive episodes''' '''Insights''' '''🔗 The missing link:''' * OGM persists '''after depression''' * OGM predicts '''future depressive symptoms''' '''👉 So logically:''' * OGM is not just a byproduct * It’s a '''stable vulnerability mechanism''' We can argue: OGM = vulnerability → (depression) → suicidal ideation. === Watkins et. al (2020) === * Year: '''2000 (older, but foundational)''' * Type: '''Experimental study''' * Focus: Can OGM be '''changed/manipulated in real time?''' '''Conclusions''' '''1. Distraction ↓ OGM''' * Less overgeneral memory when people stop ruminating ----'''2. Decentring ↓ OGM''' * Thinking more flexibly → more '''specific memories''' ----'''3. Rumination ↑/maintains OGM''' * Overthinking keeps memory vague ----'''🧠 Big takeaway:'''<blockquote>OGM is '''not fixed''' — it depends on your '''current cognitive state'''</blockquote>...could we argue: * '''Rumination → OGM''' * '''OGM → poor problem solving / hopelessness''' * '''→ Suicidal ideation''' '''Limitations''' * ❌ Small sample (N=48) * ❌ Experimental / lab setting * ❌ Not about suicidal ideation directly * ❌ Older study === Young et. al (2013) – OGM in MDD & high-risk individuals (fMRI study) === * Sample: ** MDD patients ** High-risk individuals (family history) ** Healthy controls * Used '''autobiographical memory task + fMRI''' '''Behavioral findings''' * Both '''MDD and high-risk groups''': ** '''Fewer specific memories''' ** '''More overgeneral (categorical) memories''' * High-risk group ≈ MDD group (no major difference) 👉 Huge point:<blockquote>OGM exists '''before depression fully develops'''</blockquote>'''Insights''' * Differences in: ** '''medial prefrontal cortex''' ** '''anterior cingulate cortex (ACC)''' ** '''occipital regions''' 👉 These areas are tied to: * self-referential thinking * emotion processing * rumination ("[https://pmc.ncbi.nlm.nih.gov/articles/PMC3794427/#S3 there nonetheless seems to be a general convergence on the anterior cortical midline structures as playing an important role in maladaptive rumination in major depression]") * '''OGM''' appears to exist in people that don't have depression, so OGM is a risk factor. * Observes people who ''become'' depressed, not just depressed people. * OGM is linked to brain systems involved in self-focus and rumination. That connects directly to: ** CaR-FA-X ** hopelessness ** suicidal ideation pathways '''Limitations''' * Super small sample (n=16) * Neuroimaging, not suicide-oriented. === Stange et al. (2013) – OGM × emotional maltreatment → depression === * Longitudinal adolescent sample (N=174, age ~12–13) * Tested:<blockquote>'''Does OGM interact with stress (emotional abuse/neglect) to predict depression?'''</blockquote> '''Findings''' 1. OGM alone ≠ strong predictor * OGM '''by itself did not strongly predict depression''' ----2. OGM × emotional abuse → ↑ depression * Among adolescents with '''high emotional abuse''': ** More OGM → '''greater increases in depressive symptoms''' * Among low-stress individuals: ** OGM had little effect ----3. Key interpretation:<blockquote>OGM is a '''vulnerability factor that requires stress to activate'''</blockquote> ----4. Mechanism support Paper explicitly links OGM to: * rumination * avoidance * poor problem solving '''Insights''' * '''A clean diathesis-stress model is present''': explaining the connection between OGM and SI, while explaining why OGM is not a predictor [as seen in this study and the insights amongst low-stress individuals]. * '''🔗 Integration:''' * Crane (2016):<blockquote>OGM doesn’t predict SI in general population</blockquote> * Stange (2013):<blockquote>OGM only matters when '''stress is present'''</blockquote> 👉 That resolves the contradiction What could we get from this? "OGM may not independently predict psychopathology but interacts with stressors (e.g., emotional maltreatment) to increase vulnerability." == Narratives? == == See also == * [[wikipedia:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]] == Notes == {{Reflist}} byadxxyd3r6dxm7okch2jbjjkrdf4an 2807030 2807029 2026-04-29T17:03:57Z Atcovi 276019 upd. 2807030 wikitext text/x-wiki ''Paper count:'' 15 (4/29/2026) == Research == === [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] === '''Background''' * Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner<ref>The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.</ref>. * Mentions number of studies associating OGM with suicide: "''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''" * [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓ * '''Research question(s)?''' No clear evidence to show how childhood trauma & OGM interact in the suicidal process and whether depression is a ''moderating'' effect. * '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals. * '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways. '''Method''' * '''356 Chinese participants'''. ** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. ** '''176 healthy individuals''' '''Measures''' * '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. * '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. * '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. * '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38. ** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. ** ↑ BSI-CV score ↑ SI * '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: "how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?". '''Results''' * Trauma → OGM, WSI, CSI * OGM → WSI + CSI * WSI → CSI * WSI → PSB * PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]''' WSI is strongly predictive of: * PSB * future SI intensity '''Discussion''' # '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically. # '''WSI''' = worst suicidal thoughts one has ever had [at a certain point]. # '''CSI''' = current point of suicidal ideation * [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together. * '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. ** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI. ** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. <-- appears to be relevant only with other paired vulnerabilities (ex, trauma). ** OGM as a stable trait of depression was more severe in the depression group vs. healthy control. ** Childhood trauma & OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model. ** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity. ** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation. ** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) & WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation. * Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy & MeST, both geared towards recalling memories in detail. * '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors. * '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior. === [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] === Crane et al. (2016) – Community adolescent cohort * Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16) * Tested whether OGM predicts depression/suicidality and moderates life stress * '''Findings:''' ** OGM did not predict depression, suicidal ideation, or self-harm ** OGM did not moderate the effect of life events ** Life events strongly predicted all outcomes * '''Interpretation:''' ** OGM may not function as a general vulnerability factor in community samples ** Likely only relevant in high-risk or depressed populations === Sumner et al. (2010) – Meta-analysis (OGM & depression) === * Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression * '''Findings:''' ** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up ** Effect remains '''even after controlling for baseline depression''' ** Overall effect size is '''small but significant (~1–2% variance explained)''' * '''Moderators:''' ** Stronger effect in '''clinically depressed samples vs nonclinical''' ** Stronger with '''shorter follow-up periods''' * '''Interpretation:''' ** OGM is a '''predictor of depression maintenance''', not just a correlate ** Likely acts as a '''vulnerability factor''', especially in high-risk groups {{Notice|OGM works under certain conditions (clinical / high-risk), not universally}} === Moore & Zoellner (2007) – Evaluative Review (OGM & trauma) === * Evaluative review of '''24 studies''' examining trauma exposure and OGM * '''Core question:''' Does trauma cause overgeneral memory? '''Findings''' * ❌ '''No consistent link''' between trauma exposure and OGM * ✅ OGM is more consistently linked to: ** '''Depression''' ** '''PTSD symptoms (intrusions, avoidance)''' * Trauma exposure alone is '''not sufficient''' to produce OGM * OGM appears more tied to '''psychopathology''', not the event itself '''Key nuance''' * Post-trauma '''symptoms''' (not the trauma itself) are what matter * Evidence across studies is '''mixed and methodologically inconsistent''' '''Interpretation''' * Challenges the classic “trauma → OGM” theory * Supports alternative view:<blockquote>OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)</blockquote> * Trauma alone isn’t enough — psychological response matters === Crane & Duggan (2009) – CSA onset & OGM in suicidal patients === * Clinical sample of '''49 patients with recurrent suicidal behavior''' * Examined whether '''age of onset of childhood sexual abuse (CSA)''' relates to OGM '''Findings''' * Earlier CSA onset → '''greater OGM''' (fewer specific, more categorical memories) * Effect remained after controlling for: ** depression ** verbal fluency * Presence vs absence of CSA '''did NOT differ in OGM''' ** (because sample already highly clinical) '''Interpretation''' * Not trauma itself → but '''timing of trauma (early development)''' matters * Supports idea that OGM develops when: ** memory systems are still forming ** avoidance becomes a learned coping style '''Insights''' # Analysis of OGM in a suicidal population. # Moore & Zoellner had trauma alone ≠ OGM, while Crane & Duggan added some trauma characteristics from early developmental trauma. # ''To take note'': Earlier trauma is associated with greater OGM '''within high-risk groups.''' === Champagne et al. (2016) – OGM in adolescent MDD === * Clinical adolescent sample (ages 11–18; current MDD, remitted MDD, never depressed) * Tested whether OGM is '''state (only during depression)''' or '''trait (persists after)''' '''Findings:''' * Adolescents with '''current AND remitted MDD''' showed more OGM than controls * No significant difference between current vs remitted groups * OGM present for both '''positive and negative cues''' '''Interpretation:''' * OGM is likely a '''trait-like vulnerability''', not just a temporary symptom * May contribute to '''risk of recurrence''' in depression '''1. Trait vs state question (very important)''' This paper directly answers:<blockquote>Is OGM just a symptom or a vulnerability?</blockquote>👉 Answer: '''likely vulnerability''' That’s big for your argument. ----'''2. Adolescent + clinical sample''' * Not just adults * Not just community 👉 Stronger relevance than many papers ----'''3. Fits your mechanism pathway''' Supports:<blockquote>OGM persists → increases long-term risk → can feed into suicidal ideation</blockquote>'''Limitations''' * Small sample (n=65 total) * Cross-sectional → can’t prove causality * Can’t tell if OGM came '''before''' depression or is a “scar” === Arie et al. (2008) – OGM, problem solving & suicidal behavior === * Clinical adolescent inpatient sample (suicidal vs nonsuicidal vs healthy controls) * Tested Williams’ model linking OGM, problem solving, and suicide '''Findings:''' * Suicidal adolescents showed: ** '''More OGM (less specific memory)''' ** '''Worse interpersonal problem-solving ability''' ** '''Higher hopelessness''' * OGM significantly correlated with: ** poorer problem solving ** higher hopelessness ** greater suicidal behavior * Negative childhood life events also linked to OGM and suicide '''Interpretation:''' * Supports pathway:<blockquote>OGM → poor problem solving → hopelessness → suicidal behavior</blockquote> * OGM may limit access to specific past experiences needed to solve problems effectively '''Insights''' * Study on population actually attempting suicide. * Points out a clear pathway * Suicidal group had lowest memory specificity, worst problem-solving abilities, and highest hopelessness '''Limitations''' * n=75, small sample * cross sectional, so causality can not be pinpointed * inpatient sample, so cannot be generalizable === Kaviani et al. (2011) – OGM, problem solving & suicidal ideation === * Clinical sample: depressed patients '''with vs without suicidal ideation''' * Tested links between OGM, problem solving, depression, and hopelessness '''Findings:''' * Suicidal ideation group: ** '''Less specific autobiographical memories (more OGM)''' ** '''More hopelessness''' ** '''Less effective problem-solving strategies''' * OGM associated with: ** poorer problem solving ** greater hopelessness * Evidence for a “cycle”:<blockquote>OGM → poor problem solving → hopelessness → suicidal ideation</blockquote> '''Interpretation:''' * OGM may contribute to suicidal ideation by '''impairing cognitive functioning''', especially problem solving * Supports a '''cognitive pathway to suicide risk''' '''Insights''' * Studies suicidal ideation directly. * Mechanism displayed: OGM --> problem solving --> hopelessness --> suicidal ideation * Within-depression comparison '''Limitations''' * n=40 * Cross-sectional, lacks causality * Specific country (Iran) === Zhu et al. (2025) – OGM & suicidal ideation (ML study) === * Clinical sample (n=88 depressed patients; with vs without suicidal ideation across severity levels) * Used '''AMT + machine learning + vocal features''' to distinguish suicidal ideation '''Findings:''' * Patients with suicidal ideation: ** '''More OGM (fewer specific memories)''' ** Especially impaired recall of '''positive memories''' * OGM strongly associated with '''severity of suicidal ideation''' * Crucially: ** '''OGM predicts suicidal ideation independently of depression severity''' * ML model showed: ** OGM features were key for identifying '''presence of suicidal ideation''' ** Depression severity features ≠ sufficient to predict SI '''Interpretation:''' * OGM is a '''specific cognitive marker of suicidal ideation''', not just a byproduct of depression * Supports:<blockquote>OGM contributes uniquely to suicidal ideation beyond general depression</blockquote>'''Insights''' * OGM is unique in contribution/specifically tied to SI. * Separated depression anxiety and suicidal ideation. OGM is not merely a correlate of depression, but a distinct cognitive vulnerability that contributes specifically to suicidal ideation. === Weiss-Cowie et al. (2023) — Meta-analysis on OGM & depression === '''🔑 Core findings (clean summary)''' * Massive meta-analysis ('''67 studies''') * Depressed individuals show: ** '''Less specific memories''' → ''(g = −0.73)'' ** '''More overgeneral memories''' → ''(g = 0.77)'' 👉 That’s '''medium–large effect sizes''' → very strong evidence ----'''🧠 Important conceptual findings''' '''1. OGM exists across:''' * Current depression * Subthreshold depression * Remitted depression 👉 Translation:<blockquote>OGM is not just a “currently depressed” phenomenon</blockquote> ----'''2. Trait vs state (VERY important for your work)''' * OGM persists even after remission * But is '''stronger during active depression''' 👉 This supports:<blockquote>OGM = '''trait vulnerability that gets amplified during depression'''</blockquote> ----'''3. Supports CaR-FA-X model''' * Capture & rumination * Functional avoidance * Executive dysfunction 👉 This is your '''mechanism backbone paper''' === Crane et. al (2016) === * Year: '''2016''' (so not “new,” but still solid) * Design: '''Large longitudinal cohort (ALSPAC)''' * Sample: ~5,700 adolescents * Timeline: ** OGM at age 13 ** Outcomes at age 16 '''Conclusions'''<blockquote>'''No meaningful relationship between OGM and later:'''</blockquote> * Depression * Suicidal ideation * Self-harm After controlling for confounds. This paper is basically saying:<blockquote>“OGM is NOT a general population risk factor for later psychopathology.”</blockquote>More precisely: * Works in: ** Clinical samples ** High-risk groups * '''Does NOT generalize to community samples''' '''Insights''' * high-quality longitudinal data '''1. Show inconsistency'''<blockquote>“While some studies show associations between OGM and suicidality, large community-based longitudinal evidence suggests no such relationship.”</blockquote> ----'''2. Introduce moderation / context argument'''<blockquote>“These discrepancies suggest that OGM may function as a vulnerability factor only under specific conditions (e.g., clinical populations, high-risk individuals).”</blockquote> ----'''3. Strengthen your theoretical argument''' This paper supports: * '''CaR-FA-X conditional vulnerability''' * '''Diathesis-stress framing''' === Hallford et. al (2022) === * Year: '''2022 → recent''' * Type: '''Systematic review + meta-analysis''' * Focus: OGM in '''remitted depression''' * Sample: '''17 studies''' '''Conclusion''' * People with '''remitted depression''': ** Recall '''fewer specific memories''' (g ≈ -0.31) ** Recall '''more overgeneral/categoric memories''' (g ≈ +0.25) 👉 These are '''small–moderate effects''', but consistent. ----<blockquote>OGM '''persists even after depression is gone'''</blockquote>This is HUGE. ----<blockquote>OGM may be a '''trait-like cognitive vulnerability''', not just a symptom</blockquote>The paper explicitly frames it as: * A '''risk factor for future depressive episodes''' '''Insights''' '''🔗 The missing link:''' * OGM persists '''after depression''' * OGM predicts '''future depressive symptoms''' '''👉 So logically:''' * OGM is not just a byproduct * It’s a '''stable vulnerability mechanism''' We can argue: OGM = vulnerability → (depression) → suicidal ideation. === Watkins et. al (2020) === * Year: '''2000 (older, but foundational)''' * Type: '''Experimental study''' * Focus: Can OGM be '''changed/manipulated in real time?''' '''Conclusions''' '''1. Distraction ↓ OGM''' * Less overgeneral memory when people stop ruminating ----'''2. Decentring ↓ OGM''' * Thinking more flexibly → more '''specific memories''' ----'''3. Rumination ↑/maintains OGM''' * Overthinking keeps memory vague ----'''🧠 Big takeaway:'''<blockquote>OGM is '''not fixed''' — it depends on your '''current cognitive state'''</blockquote>...could we argue: * '''Rumination → OGM''' * '''OGM → poor problem solving / hopelessness''' * '''→ Suicidal ideation''' '''Limitations''' * ❌ Small sample (N=48) * ❌ Experimental / lab setting * ❌ Not about suicidal ideation directly * ❌ Older study === Young et. al (2013) – OGM in MDD & high-risk individuals (fMRI study) === * Sample: ** MDD patients ** High-risk individuals (family history) ** Healthy controls * Used '''autobiographical memory task + fMRI''' '''Behavioral findings''' * Both '''MDD and high-risk groups''': ** '''Fewer specific memories''' ** '''More overgeneral (categorical) memories''' * High-risk group ≈ MDD group (no major difference) 👉 Huge point:<blockquote>OGM exists '''before depression fully develops'''</blockquote>'''Insights''' * Differences in: ** '''medial prefrontal cortex''' ** '''anterior cingulate cortex (ACC)''' ** '''occipital regions''' 👉 These areas are tied to: * self-referential thinking * emotion processing * rumination ("[https://pmc.ncbi.nlm.nih.gov/articles/PMC3794427/#S3 there nonetheless seems to be a general convergence on the anterior cortical midline structures as playing an important role in maladaptive rumination in major depression]") * '''OGM''' appears to exist in people that don't have depression, so OGM is a risk factor. * Observes people who ''become'' depressed, not just depressed people. * OGM is linked to brain systems involved in self-focus and rumination. That connects directly to: ** CaR-FA-X ** hopelessness ** suicidal ideation pathways '''Limitations''' * Super small sample (n=16) * Neuroimaging, not suicide-oriented. === Stange et al. (2013) – OGM × emotional maltreatment → depression === * Longitudinal adolescent sample (N=174, age ~12–13) * Tested:<blockquote>'''Does OGM interact with stress (emotional abuse/neglect) to predict depression?'''</blockquote> '''Findings''' 1. OGM alone ≠ strong predictor * OGM '''by itself did not strongly predict depression''' ----2. OGM × emotional abuse → ↑ depression * Among adolescents with '''high emotional abuse''': ** More OGM → '''greater increases in depressive symptoms''' * Among low-stress individuals: ** OGM had little effect ----3. Key interpretation:<blockquote>OGM is a '''vulnerability factor that requires stress to activate'''</blockquote> ----4. Mechanism support Paper explicitly links OGM to: * rumination * avoidance * poor problem solving '''Insights''' * '''A clean diathesis-stress model is present''': explaining the connection between OGM and SI, while explaining why OGM is not a predictor [as seen in this study and the insights amongst low-stress individuals]. * '''🔗 Integration:''' * Crane (2016):<blockquote>OGM doesn’t predict SI in general population</blockquote> * Stange (2013):<blockquote>OGM only matters when '''stress is present'''</blockquote> 👉 That resolves the contradiction What could we get from this? "OGM may not independently predict psychopathology but interacts with stressors (e.g., emotional maltreatment) to increase vulnerability." == Narratives? == == See also == * [[wikipedia:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]] == Notes == {{Reflist}} s90zjptuvmcpwk3gg47fntgowmoi6h0 2807031 2807030 2026-04-29T17:04:23Z Atcovi 276019 +cat 2807031 wikitext text/x-wiki ''Paper count:'' 15 (4/29/2026) == Research == === [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] === '''Background''' * Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner<ref>The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.</ref>. * Mentions number of studies associating OGM with suicide: "''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''" * [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓ * '''Research question(s)?''' No clear evidence to show how childhood trauma & OGM interact in the suicidal process and whether depression is a ''moderating'' effect. * '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals. * '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways. '''Method''' * '''356 Chinese participants'''. ** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. ** '''176 healthy individuals''' '''Measures''' * '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. * '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. * '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. * '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38. ** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. ** ↑ BSI-CV score ↑ SI * '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: "how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?". '''Results''' * Trauma → OGM, WSI, CSI * OGM → WSI + CSI * WSI → CSI * WSI → PSB * PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]''' WSI is strongly predictive of: * PSB * future SI intensity '''Discussion''' # '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically. # '''WSI''' = worst suicidal thoughts one has ever had [at a certain point]. # '''CSI''' = current point of suicidal ideation * [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together. * '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. ** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI. ** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. <-- appears to be relevant only with other paired vulnerabilities (ex, trauma). ** OGM as a stable trait of depression was more severe in the depression group vs. healthy control. ** Childhood trauma & OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model. ** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity. ** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation. ** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) & WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation. * Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy & MeST, both geared towards recalling memories in detail. * '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors. * '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior. === [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] === Crane et al. (2016) – Community adolescent cohort * Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16) * Tested whether OGM predicts depression/suicidality and moderates life stress * '''Findings:''' ** OGM did not predict depression, suicidal ideation, or self-harm ** OGM did not moderate the effect of life events ** Life events strongly predicted all outcomes * '''Interpretation:''' ** OGM may not function as a general vulnerability factor in community samples ** Likely only relevant in high-risk or depressed populations === Sumner et al. (2010) – Meta-analysis (OGM & depression) === * Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression * '''Findings:''' ** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up ** Effect remains '''even after controlling for baseline depression''' ** Overall effect size is '''small but significant (~1–2% variance explained)''' * '''Moderators:''' ** Stronger effect in '''clinically depressed samples vs nonclinical''' ** Stronger with '''shorter follow-up periods''' * '''Interpretation:''' ** OGM is a '''predictor of depression maintenance''', not just a correlate ** Likely acts as a '''vulnerability factor''', especially in high-risk groups {{Notice|OGM works under certain conditions (clinical / high-risk), not universally}} === Moore & Zoellner (2007) – Evaluative Review (OGM & trauma) === * Evaluative review of '''24 studies''' examining trauma exposure and OGM * '''Core question:''' Does trauma cause overgeneral memory? '''Findings''' * ❌ '''No consistent link''' between trauma exposure and OGM * ✅ OGM is more consistently linked to: ** '''Depression''' ** '''PTSD symptoms (intrusions, avoidance)''' * Trauma exposure alone is '''not sufficient''' to produce OGM * OGM appears more tied to '''psychopathology''', not the event itself '''Key nuance''' * Post-trauma '''symptoms''' (not the trauma itself) are what matter * Evidence across studies is '''mixed and methodologically inconsistent''' '''Interpretation''' * Challenges the classic “trauma → OGM” theory * Supports alternative view:<blockquote>OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)</blockquote> * Trauma alone isn’t enough — psychological response matters === Crane & Duggan (2009) – CSA onset & OGM in suicidal patients === * Clinical sample of '''49 patients with recurrent suicidal behavior''' * Examined whether '''age of onset of childhood sexual abuse (CSA)''' relates to OGM '''Findings''' * Earlier CSA onset → '''greater OGM''' (fewer specific, more categorical memories) * Effect remained after controlling for: ** depression ** verbal fluency * Presence vs absence of CSA '''did NOT differ in OGM''' ** (because sample already highly clinical) '''Interpretation''' * Not trauma itself → but '''timing of trauma (early development)''' matters * Supports idea that OGM develops when: ** memory systems are still forming ** avoidance becomes a learned coping style '''Insights''' # Analysis of OGM in a suicidal population. # Moore & Zoellner had trauma alone ≠ OGM, while Crane & Duggan added some trauma characteristics from early developmental trauma. # ''To take note'': Earlier trauma is associated with greater OGM '''within high-risk groups.''' === Champagne et al. (2016) – OGM in adolescent MDD === * Clinical adolescent sample (ages 11–18; current MDD, remitted MDD, never depressed) * Tested whether OGM is '''state (only during depression)''' or '''trait (persists after)''' '''Findings:''' * Adolescents with '''current AND remitted MDD''' showed more OGM than controls * No significant difference between current vs remitted groups * OGM present for both '''positive and negative cues''' '''Interpretation:''' * OGM is likely a '''trait-like vulnerability''', not just a temporary symptom * May contribute to '''risk of recurrence''' in depression '''1. Trait vs state question (very important)''' This paper directly answers:<blockquote>Is OGM just a symptom or a vulnerability?</blockquote>👉 Answer: '''likely vulnerability''' That’s big for your argument. ----'''2. Adolescent + clinical sample''' * Not just adults * Not just community 👉 Stronger relevance than many papers ----'''3. Fits your mechanism pathway''' Supports:<blockquote>OGM persists → increases long-term risk → can feed into suicidal ideation</blockquote>'''Limitations''' * Small sample (n=65 total) * Cross-sectional → can’t prove causality * Can’t tell if OGM came '''before''' depression or is a “scar” === Arie et al. (2008) – OGM, problem solving & suicidal behavior === * Clinical adolescent inpatient sample (suicidal vs nonsuicidal vs healthy controls) * Tested Williams’ model linking OGM, problem solving, and suicide '''Findings:''' * Suicidal adolescents showed: ** '''More OGM (less specific memory)''' ** '''Worse interpersonal problem-solving ability''' ** '''Higher hopelessness''' * OGM significantly correlated with: ** poorer problem solving ** higher hopelessness ** greater suicidal behavior * Negative childhood life events also linked to OGM and suicide '''Interpretation:''' * Supports pathway:<blockquote>OGM → poor problem solving → hopelessness → suicidal behavior</blockquote> * OGM may limit access to specific past experiences needed to solve problems effectively '''Insights''' * Study on population actually attempting suicide. * Points out a clear pathway * Suicidal group had lowest memory specificity, worst problem-solving abilities, and highest hopelessness '''Limitations''' * n=75, small sample * cross sectional, so causality can not be pinpointed * inpatient sample, so cannot be generalizable === Kaviani et al. (2011) – OGM, problem solving & suicidal ideation === * Clinical sample: depressed patients '''with vs without suicidal ideation''' * Tested links between OGM, problem solving, depression, and hopelessness '''Findings:''' * Suicidal ideation group: ** '''Less specific autobiographical memories (more OGM)''' ** '''More hopelessness''' ** '''Less effective problem-solving strategies''' * OGM associated with: ** poorer problem solving ** greater hopelessness * Evidence for a “cycle”:<blockquote>OGM → poor problem solving → hopelessness → suicidal ideation</blockquote> '''Interpretation:''' * OGM may contribute to suicidal ideation by '''impairing cognitive functioning''', especially problem solving * Supports a '''cognitive pathway to suicide risk''' '''Insights''' * Studies suicidal ideation directly. * Mechanism displayed: OGM --> problem solving --> hopelessness --> suicidal ideation * Within-depression comparison '''Limitations''' * n=40 * Cross-sectional, lacks causality * Specific country (Iran) === Zhu et al. (2025) – OGM & suicidal ideation (ML study) === * Clinical sample (n=88 depressed patients; with vs without suicidal ideation across severity levels) * Used '''AMT + machine learning + vocal features''' to distinguish suicidal ideation '''Findings:''' * Patients with suicidal ideation: ** '''More OGM (fewer specific memories)''' ** Especially impaired recall of '''positive memories''' * OGM strongly associated with '''severity of suicidal ideation''' * Crucially: ** '''OGM predicts suicidal ideation independently of depression severity''' * ML model showed: ** OGM features were key for identifying '''presence of suicidal ideation''' ** Depression severity features ≠ sufficient to predict SI '''Interpretation:''' * OGM is a '''specific cognitive marker of suicidal ideation''', not just a byproduct of depression * Supports:<blockquote>OGM contributes uniquely to suicidal ideation beyond general depression</blockquote>'''Insights''' * OGM is unique in contribution/specifically tied to SI. * Separated depression anxiety and suicidal ideation. OGM is not merely a correlate of depression, but a distinct cognitive vulnerability that contributes specifically to suicidal ideation. === Weiss-Cowie et al. (2023) — Meta-analysis on OGM & depression === '''🔑 Core findings (clean summary)''' * Massive meta-analysis ('''67 studies''') * Depressed individuals show: ** '''Less specific memories''' → ''(g = −0.73)'' ** '''More overgeneral memories''' → ''(g = 0.77)'' 👉 That’s '''medium–large effect sizes''' → very strong evidence ----'''🧠 Important conceptual findings''' '''1. OGM exists across:''' * Current depression * Subthreshold depression * Remitted depression 👉 Translation:<blockquote>OGM is not just a “currently depressed” phenomenon</blockquote> ----'''2. Trait vs state (VERY important for your work)''' * OGM persists even after remission * But is '''stronger during active depression''' 👉 This supports:<blockquote>OGM = '''trait vulnerability that gets amplified during depression'''</blockquote> ----'''3. Supports CaR-FA-X model''' * Capture & rumination * Functional avoidance * Executive dysfunction 👉 This is your '''mechanism backbone paper''' === Crane et. al (2016) === * Year: '''2016''' (so not “new,” but still solid) * Design: '''Large longitudinal cohort (ALSPAC)''' * Sample: ~5,700 adolescents * Timeline: ** OGM at age 13 ** Outcomes at age 16 '''Conclusions'''<blockquote>'''No meaningful relationship between OGM and later:'''</blockquote> * Depression * Suicidal ideation * Self-harm After controlling for confounds. This paper is basically saying:<blockquote>“OGM is NOT a general population risk factor for later psychopathology.”</blockquote>More precisely: * Works in: ** Clinical samples ** High-risk groups * '''Does NOT generalize to community samples''' '''Insights''' * high-quality longitudinal data '''1. Show inconsistency'''<blockquote>“While some studies show associations between OGM and suicidality, large community-based longitudinal evidence suggests no such relationship.”</blockquote> ----'''2. Introduce moderation / context argument'''<blockquote>“These discrepancies suggest that OGM may function as a vulnerability factor only under specific conditions (e.g., clinical populations, high-risk individuals).”</blockquote> ----'''3. Strengthen your theoretical argument''' This paper supports: * '''CaR-FA-X conditional vulnerability''' * '''Diathesis-stress framing''' === Hallford et. al (2022) === * Year: '''2022 → recent''' * Type: '''Systematic review + meta-analysis''' * Focus: OGM in '''remitted depression''' * Sample: '''17 studies''' '''Conclusion''' * People with '''remitted depression''': ** Recall '''fewer specific memories''' (g ≈ -0.31) ** Recall '''more overgeneral/categoric memories''' (g ≈ +0.25) 👉 These are '''small–moderate effects''', but consistent. ----<blockquote>OGM '''persists even after depression is gone'''</blockquote>This is HUGE. ----<blockquote>OGM may be a '''trait-like cognitive vulnerability''', not just a symptom</blockquote>The paper explicitly frames it as: * A '''risk factor for future depressive episodes''' '''Insights''' '''🔗 The missing link:''' * OGM persists '''after depression''' * OGM predicts '''future depressive symptoms''' '''👉 So logically:''' * OGM is not just a byproduct * It’s a '''stable vulnerability mechanism''' We can argue: OGM = vulnerability → (depression) → suicidal ideation. === Watkins et. al (2020) === * Year: '''2000 (older, but foundational)''' * Type: '''Experimental study''' * Focus: Can OGM be '''changed/manipulated in real time?''' '''Conclusions''' '''1. Distraction ↓ OGM''' * Less overgeneral memory when people stop ruminating ----'''2. Decentring ↓ OGM''' * Thinking more flexibly → more '''specific memories''' ----'''3. Rumination ↑/maintains OGM''' * Overthinking keeps memory vague ----'''🧠 Big takeaway:'''<blockquote>OGM is '''not fixed''' — it depends on your '''current cognitive state'''</blockquote>...could we argue: * '''Rumination → OGM''' * '''OGM → poor problem solving / hopelessness''' * '''→ Suicidal ideation''' '''Limitations''' * ❌ Small sample (N=48) * ❌ Experimental / lab setting * ❌ Not about suicidal ideation directly * ❌ Older study === Young et. al (2013) – OGM in MDD & high-risk individuals (fMRI study) === * Sample: ** MDD patients ** High-risk individuals (family history) ** Healthy controls * Used '''autobiographical memory task + fMRI''' '''Behavioral findings''' * Both '''MDD and high-risk groups''': ** '''Fewer specific memories''' ** '''More overgeneral (categorical) memories''' * High-risk group ≈ MDD group (no major difference) 👉 Huge point:<blockquote>OGM exists '''before depression fully develops'''</blockquote>'''Insights''' * Differences in: ** '''medial prefrontal cortex''' ** '''anterior cingulate cortex (ACC)''' ** '''occipital regions''' 👉 These areas are tied to: * self-referential thinking * emotion processing * rumination ("[https://pmc.ncbi.nlm.nih.gov/articles/PMC3794427/#S3 there nonetheless seems to be a general convergence on the anterior cortical midline structures as playing an important role in maladaptive rumination in major depression]") * '''OGM''' appears to exist in people that don't have depression, so OGM is a risk factor. * Observes people who ''become'' depressed, not just depressed people. * OGM is linked to brain systems involved in self-focus and rumination. That connects directly to: ** CaR-FA-X ** hopelessness ** suicidal ideation pathways '''Limitations''' * Super small sample (n=16) * Neuroimaging, not suicide-oriented. === Stange et al. (2013) – OGM × emotional maltreatment → depression === * Longitudinal adolescent sample (N=174, age ~12–13) * Tested:<blockquote>'''Does OGM interact with stress (emotional abuse/neglect) to predict depression?'''</blockquote> '''Findings''' 1. OGM alone ≠ strong predictor * OGM '''by itself did not strongly predict depression''' ----2. OGM × emotional abuse → ↑ depression * Among adolescents with '''high emotional abuse''': ** More OGM → '''greater increases in depressive symptoms''' * Among low-stress individuals: ** OGM had little effect ----3. Key interpretation:<blockquote>OGM is a '''vulnerability factor that requires stress to activate'''</blockquote> ----4. Mechanism support Paper explicitly links OGM to: * rumination * avoidance * poor problem solving '''Insights''' * '''A clean diathesis-stress model is present''': explaining the connection between OGM and SI, while explaining why OGM is not a predictor [as seen in this study and the insights amongst low-stress individuals]. * '''🔗 Integration:''' * Crane (2016):<blockquote>OGM doesn’t predict SI in general population</blockquote> * Stange (2013):<blockquote>OGM only matters when '''stress is present'''</blockquote> 👉 That resolves the contradiction What could we get from this? "OGM may not independently predict psychopathology but interacts with stressors (e.g., emotional maltreatment) to increase vulnerability." == Narratives? == == See also == * [[wikipedia:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]] == Notes == {{Reflist}} [[Category:Atcovi/OGM & Suicide Poster]] jc25bljd1hjdm6v4azvmgee59e8di4w 2807034 2807031 2026-04-29T17:07:47Z Atcovi 276019 /* Sumner et al. (2010) – Meta-analysis (OGM & depression) */ https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/ 2807034 wikitext text/x-wiki ''Paper count:'' 15 (4/29/2026) == Research == === [https://link.springer.com/article/10.1186/s12888-020-02877-6 Jiang, W., Hu, G., Zhang, J. ''et al.'' Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people. ''BMC Psychiatry'' '''20''', 501 (2020). https://doi.org/10.1186/s12888-020-02877-6] === '''Background''' * Mentions [https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/ the integrated motivational–volitional model of suicidal behaviour] by O'Conner<ref>The '''integrated motivational-volitional (IMV) model of suicidal behavior''', dividing it into three phases: pre-motivational, motivational, and volitional. Firstly, the pre-motivational phase is composed of diathesis, environment, and life events, describing the background factors and triggering events. Secondly, the motivational phase focuses on the psychological processes of suicidal ideation and intent. Finally, the volitional phase governs the transition from suicidal ideation to suicide attempts.</ref>. * Mentions number of studies associating OGM with suicide: "''...it could be inferred that OGM mediates the suicidal process by preventing the individuals from solving problems and envisioning the future by searching in the past experience, thereby creating a feeling of hopelessness and helplessness.''" * [https://www.sciencedirect.com/science/article/abs/pii/S2352250X21000129 MeST] ↑ generalization of autobiographical memory ↓ * '''Research question(s)?''' No clear evidence to show how childhood trauma & OGM interact in the suicidal process and whether depression is a ''moderating'' effect. * '''Hypothesis?''' OGM has different effects on the suicide process in depression patients and healthy individuals. * '''Purpose of study?''' Aimed to compare childhood trauma, OGM, suicidal ideation, and suicidal behavior between depression patients and healthy individuals, and explore the differences caused by depression in suicidal pathways. '''Method''' * '''356 Chinese participants'''. ** '''180 depressed participants''': (''n'' = 121) 67.2% of the depressed patients were depressed for more than a year. ** '''176 healthy individuals''' '''Measures''' * '''Depressive symptoms''': BDI-II [Beck Depression Inventory-II]; 21-item self-report survey that asses depression severity symptoms for past 2 weeks, using a four point Likert scale of 0-3. * '''OGM''': OGMQ; 19-item self-report tool that assesses the specificity of autobiographical memory using a four-point Likert scale (1 = perfect math; 4 = not a match). Total scores range from 19-76, with higher the score, the more frequent general, non-specific memories come into play. * '''Childhood trauma''': [CTQ-SF]; 28-item self-report survey measuring maltreatment and trauma experience before age 16 using a 5-point Likert scale (1 = never; 5 = always). The five subscales include sexual abuse, physical abuse, emotional abuse, emotional neglect, and physical neglect. * '''Suicidal ideation''': [BSI-CV]; 19-item self-report questionnaire that evaluates the thoughts about life and death and the severity of SI, using a three-point scale of 0-2. Range: 0-38. ** If the score of item 4 or 5 is NOT 0, it indicates suicidal ideation. ** ↑ BSI-CV score ↑ SI * '''PSB:''' ...or previous suicidal behavior; 4-point scale was used (0 = never, 1 = once; 2 = twice; 3 = more than twice). Question asked was: "how many times did you induce self-injury or suicidal behaviors, such as taking medicine or cutting your wrists in the past?". '''Results''' * Trauma → OGM, WSI, CSI * OGM → WSI + CSI * WSI → CSI * WSI → PSB * PSB → CSI (reverse influence) [suggests '''impulsive attempts ≠ ideation pathway]''' WSI is strongly predictive of: * PSB * future SI intensity '''Discussion''' # '''OGM''' = overgeneralized autobiographical memory, remembering things vaguley and not specifically. # '''WSI''' = worst suicidal thoughts one has ever had [at a certain point]. # '''CSI''' = current point of suicidal ideation * [according to the author] appears to be the first study that connects CSI and WSI of depressed and healthy control groups with suicidal behavior, childhood trauma, and autobiographical memory together. * '''Results?''' Suggests that SI and behavior od epression patients are significantly HIGHER than of healthy individuals. Background factors, such as childhood trauma, and the moderator of suicide, such as OGM, are more severe in depression patients. ** ONLY in depression patients, OGM significantly affects the CSI and acts an intermediary between childhood trauma and CSI. ** ...but NO significant effect of OGM on CSI in healthy individuals, ''indicating that OGM plays different roles in the emergence of suicidal ideation in different populations''. <-- appears to be relevant only with other paired vulnerabilities (ex, trauma). ** OGM as a stable trait of depression was more severe in the depression group vs. healthy control. ** Childhood trauma & OGM were correlated with WSI, indicating that they are critical factors of SI in accordance with the IMV model. ** PSB was strongly correlated to WSI. WSI might be an independent predictor of follow-up suicidal ideation intensity. ** Another finding was that PSB was negatively correlated with CSI and less affected by WSI in the healthy group. One reason might be that most proportion of suicides in healthy people are impulsive attempts, which do not follow the depression-hopelessness path to suicidal behavior, with lower expectations of death and suicidal ideation. ** OGM + WSI mediated '''70.28% of the total effect''' between trauma and CSI [impact of trauma on CSI does not happen ''directly'', but through OGM (messes up memory style) & WSI (makes past suicidal thoughts worse)], while 30% of it is direct between trauma and current suicidal ideation. * Improving the specificity of autobiographical memory may be an effective way to prevent suicidal ideation for depression. These could be achieved through life-review therapy & MeST, both geared towards recalling memories in detail. * '''Limitations?''' Low sample size, cross-sectional study, suicide's complex nature makes it difficult to account for all possible biological and psychological factors. * '''Conclusion?''' This study identified the different roles of OGM in the suicidal ideation of depressed and healthy people. In depression patients, it affects the CSI and WSI and mediates the CSI due to the effect of childhood trauma. In healthy people, it can only affect the WSI. As an adjustable risk factor, the autobiographical memory might be a target of intervention for suicidal ideation in depression patients. Since training in specific memory retrieval has been proven to be effective in depression, future studies should consider whether it can reduce the emergence of suicidal ideation and suicidal behavior. === [https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/pdf/pmem-24-348.pdf Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort] === Crane et al. (2016) – Community adolescent cohort * Large UK longitudinal study (n≈5800 adolescents; ages 13 → 16) * Tested whether OGM predicts depression/suicidality and moderates life stress * '''Findings:''' ** OGM did not predict depression, suicidal ideation, or self-harm ** OGM did not moderate the effect of life events ** Life events strongly predicted all outcomes * '''Interpretation:''' ** OGM may not function as a general vulnerability factor in community samples ** Likely only relevant in high-risk or depressed populations === [https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/ Sumner et al. (2010) – Meta-analysis (OGM & depression)] === * Meta-analysis of '''15 studies''' examining whether OGM predicts the course of depression * '''Findings:''' ** Higher OGM (fewer specific memories) → higher depressive symptoms at follow-up ** Effect remains '''even after controlling for baseline depression''' ** Overall effect size is '''small but significant (~1–2% variance explained)''' * '''Moderators:''' ** Stronger effect in '''clinically depressed samples vs nonclinical''' ** Stronger with '''shorter follow-up periods''' * '''Interpretation:''' ** OGM is a '''predictor of depression maintenance''', not just a correlate ** Likely acts as a '''vulnerability factor''', especially in high-risk groups {{Notice|OGM works under certain conditions (clinical / high-risk), not universally}} === Moore & Zoellner (2007) – Evaluative Review (OGM & trauma) === * Evaluative review of '''24 studies''' examining trauma exposure and OGM * '''Core question:''' Does trauma cause overgeneral memory? '''Findings''' * ❌ '''No consistent link''' between trauma exposure and OGM * ✅ OGM is more consistently linked to: ** '''Depression''' ** '''PTSD symptoms (intrusions, avoidance)''' * Trauma exposure alone is '''not sufficient''' to produce OGM * OGM appears more tied to '''psychopathology''', not the event itself '''Key nuance''' * Post-trauma '''symptoms''' (not the trauma itself) are what matter * Evidence across studies is '''mixed and methodologically inconsistent''' '''Interpretation''' * Challenges the classic “trauma → OGM” theory * Supports alternative view:<blockquote>OGM = cognitive feature of clinical disorders (e.g., MDD, PTSD)</blockquote> * Trauma alone isn’t enough — psychological response matters === Crane & Duggan (2009) – CSA onset & OGM in suicidal patients === * Clinical sample of '''49 patients with recurrent suicidal behavior''' * Examined whether '''age of onset of childhood sexual abuse (CSA)''' relates to OGM '''Findings''' * Earlier CSA onset → '''greater OGM''' (fewer specific, more categorical memories) * Effect remained after controlling for: ** depression ** verbal fluency * Presence vs absence of CSA '''did NOT differ in OGM''' ** (because sample already highly clinical) '''Interpretation''' * Not trauma itself → but '''timing of trauma (early development)''' matters * Supports idea that OGM develops when: ** memory systems are still forming ** avoidance becomes a learned coping style '''Insights''' # Analysis of OGM in a suicidal population. # Moore & Zoellner had trauma alone ≠ OGM, while Crane & Duggan added some trauma characteristics from early developmental trauma. # ''To take note'': Earlier trauma is associated with greater OGM '''within high-risk groups.''' === Champagne et al. (2016) – OGM in adolescent MDD === * Clinical adolescent sample (ages 11–18; current MDD, remitted MDD, never depressed) * Tested whether OGM is '''state (only during depression)''' or '''trait (persists after)''' '''Findings:''' * Adolescents with '''current AND remitted MDD''' showed more OGM than controls * No significant difference between current vs remitted groups * OGM present for both '''positive and negative cues''' '''Interpretation:''' * OGM is likely a '''trait-like vulnerability''', not just a temporary symptom * May contribute to '''risk of recurrence''' in depression '''1. Trait vs state question (very important)''' This paper directly answers:<blockquote>Is OGM just a symptom or a vulnerability?</blockquote>👉 Answer: '''likely vulnerability''' That’s big for your argument. ----'''2. Adolescent + clinical sample''' * Not just adults * Not just community 👉 Stronger relevance than many papers ----'''3. Fits your mechanism pathway''' Supports:<blockquote>OGM persists → increases long-term risk → can feed into suicidal ideation</blockquote>'''Limitations''' * Small sample (n=65 total) * Cross-sectional → can’t prove causality * Can’t tell if OGM came '''before''' depression or is a “scar” === Arie et al. (2008) – OGM, problem solving & suicidal behavior === * Clinical adolescent inpatient sample (suicidal vs nonsuicidal vs healthy controls) * Tested Williams’ model linking OGM, problem solving, and suicide '''Findings:''' * Suicidal adolescents showed: ** '''More OGM (less specific memory)''' ** '''Worse interpersonal problem-solving ability''' ** '''Higher hopelessness''' * OGM significantly correlated with: ** poorer problem solving ** higher hopelessness ** greater suicidal behavior * Negative childhood life events also linked to OGM and suicide '''Interpretation:''' * Supports pathway:<blockquote>OGM → poor problem solving → hopelessness → suicidal behavior</blockquote> * OGM may limit access to specific past experiences needed to solve problems effectively '''Insights''' * Study on population actually attempting suicide. * Points out a clear pathway * Suicidal group had lowest memory specificity, worst problem-solving abilities, and highest hopelessness '''Limitations''' * n=75, small sample * cross sectional, so causality can not be pinpointed * inpatient sample, so cannot be generalizable === Kaviani et al. (2011) – OGM, problem solving & suicidal ideation === * Clinical sample: depressed patients '''with vs without suicidal ideation''' * Tested links between OGM, problem solving, depression, and hopelessness '''Findings:''' * Suicidal ideation group: ** '''Less specific autobiographical memories (more OGM)''' ** '''More hopelessness''' ** '''Less effective problem-solving strategies''' * OGM associated with: ** poorer problem solving ** greater hopelessness * Evidence for a “cycle”:<blockquote>OGM → poor problem solving → hopelessness → suicidal ideation</blockquote> '''Interpretation:''' * OGM may contribute to suicidal ideation by '''impairing cognitive functioning''', especially problem solving * Supports a '''cognitive pathway to suicide risk''' '''Insights''' * Studies suicidal ideation directly. * Mechanism displayed: OGM --> problem solving --> hopelessness --> suicidal ideation * Within-depression comparison '''Limitations''' * n=40 * Cross-sectional, lacks causality * Specific country (Iran) === Zhu et al. (2025) – OGM & suicidal ideation (ML study) === * Clinical sample (n=88 depressed patients; with vs without suicidal ideation across severity levels) * Used '''AMT + machine learning + vocal features''' to distinguish suicidal ideation '''Findings:''' * Patients with suicidal ideation: ** '''More OGM (fewer specific memories)''' ** Especially impaired recall of '''positive memories''' * OGM strongly associated with '''severity of suicidal ideation''' * Crucially: ** '''OGM predicts suicidal ideation independently of depression severity''' * ML model showed: ** OGM features were key for identifying '''presence of suicidal ideation''' ** Depression severity features ≠ sufficient to predict SI '''Interpretation:''' * OGM is a '''specific cognitive marker of suicidal ideation''', not just a byproduct of depression * Supports:<blockquote>OGM contributes uniquely to suicidal ideation beyond general depression</blockquote>'''Insights''' * OGM is unique in contribution/specifically tied to SI. * Separated depression anxiety and suicidal ideation. OGM is not merely a correlate of depression, but a distinct cognitive vulnerability that contributes specifically to suicidal ideation. === Weiss-Cowie et al. (2023) — Meta-analysis on OGM & depression === '''🔑 Core findings (clean summary)''' * Massive meta-analysis ('''67 studies''') * Depressed individuals show: ** '''Less specific memories''' → ''(g = −0.73)'' ** '''More overgeneral memories''' → ''(g = 0.77)'' 👉 That’s '''medium–large effect sizes''' → very strong evidence ----'''🧠 Important conceptual findings''' '''1. OGM exists across:''' * Current depression * Subthreshold depression * Remitted depression 👉 Translation:<blockquote>OGM is not just a “currently depressed” phenomenon</blockquote> ----'''2. Trait vs state (VERY important for your work)''' * OGM persists even after remission * But is '''stronger during active depression''' 👉 This supports:<blockquote>OGM = '''trait vulnerability that gets amplified during depression'''</blockquote> ----'''3. Supports CaR-FA-X model''' * Capture & rumination * Functional avoidance * Executive dysfunction 👉 This is your '''mechanism backbone paper''' === Crane et. al (2016) === * Year: '''2016''' (so not “new,” but still solid) * Design: '''Large longitudinal cohort (ALSPAC)''' * Sample: ~5,700 adolescents * Timeline: ** OGM at age 13 ** Outcomes at age 16 '''Conclusions'''<blockquote>'''No meaningful relationship between OGM and later:'''</blockquote> * Depression * Suicidal ideation * Self-harm After controlling for confounds. This paper is basically saying:<blockquote>“OGM is NOT a general population risk factor for later psychopathology.”</blockquote>More precisely: * Works in: ** Clinical samples ** High-risk groups * '''Does NOT generalize to community samples''' '''Insights''' * high-quality longitudinal data '''1. Show inconsistency'''<blockquote>“While some studies show associations between OGM and suicidality, large community-based longitudinal evidence suggests no such relationship.”</blockquote> ----'''2. Introduce moderation / context argument'''<blockquote>“These discrepancies suggest that OGM may function as a vulnerability factor only under specific conditions (e.g., clinical populations, high-risk individuals).”</blockquote> ----'''3. Strengthen your theoretical argument''' This paper supports: * '''CaR-FA-X conditional vulnerability''' * '''Diathesis-stress framing''' === Hallford et. al (2022) === * Year: '''2022 → recent''' * Type: '''Systematic review + meta-analysis''' * Focus: OGM in '''remitted depression''' * Sample: '''17 studies''' '''Conclusion''' * People with '''remitted depression''': ** Recall '''fewer specific memories''' (g ≈ -0.31) ** Recall '''more overgeneral/categoric memories''' (g ≈ +0.25) 👉 These are '''small–moderate effects''', but consistent. ----<blockquote>OGM '''persists even after depression is gone'''</blockquote>This is HUGE. ----<blockquote>OGM may be a '''trait-like cognitive vulnerability''', not just a symptom</blockquote>The paper explicitly frames it as: * A '''risk factor for future depressive episodes''' '''Insights''' '''🔗 The missing link:''' * OGM persists '''after depression''' * OGM predicts '''future depressive symptoms''' '''👉 So logically:''' * OGM is not just a byproduct * It’s a '''stable vulnerability mechanism''' We can argue: OGM = vulnerability → (depression) → suicidal ideation. === Watkins et. al (2020) === * Year: '''2000 (older, but foundational)''' * Type: '''Experimental study''' * Focus: Can OGM be '''changed/manipulated in real time?''' '''Conclusions''' '''1. Distraction ↓ OGM''' * Less overgeneral memory when people stop ruminating ----'''2. Decentring ↓ OGM''' * Thinking more flexibly → more '''specific memories''' ----'''3. Rumination ↑/maintains OGM''' * Overthinking keeps memory vague ----'''🧠 Big takeaway:'''<blockquote>OGM is '''not fixed''' — it depends on your '''current cognitive state'''</blockquote>...could we argue: * '''Rumination → OGM''' * '''OGM → poor problem solving / hopelessness''' * '''→ Suicidal ideation''' '''Limitations''' * ❌ Small sample (N=48) * ❌ Experimental / lab setting * ❌ Not about suicidal ideation directly * ❌ Older study === Young et. al (2013) – OGM in MDD & high-risk individuals (fMRI study) === * Sample: ** MDD patients ** High-risk individuals (family history) ** Healthy controls * Used '''autobiographical memory task + fMRI''' '''Behavioral findings''' * Both '''MDD and high-risk groups''': ** '''Fewer specific memories''' ** '''More overgeneral (categorical) memories''' * High-risk group ≈ MDD group (no major difference) 👉 Huge point:<blockquote>OGM exists '''before depression fully develops'''</blockquote>'''Insights''' * Differences in: ** '''medial prefrontal cortex''' ** '''anterior cingulate cortex (ACC)''' ** '''occipital regions''' 👉 These areas are tied to: * self-referential thinking * emotion processing * rumination ("[https://pmc.ncbi.nlm.nih.gov/articles/PMC3794427/#S3 there nonetheless seems to be a general convergence on the anterior cortical midline structures as playing an important role in maladaptive rumination in major depression]") * '''OGM''' appears to exist in people that don't have depression, so OGM is a risk factor. * Observes people who ''become'' depressed, not just depressed people. * OGM is linked to brain systems involved in self-focus and rumination. That connects directly to: ** CaR-FA-X ** hopelessness ** suicidal ideation pathways '''Limitations''' * Super small sample (n=16) * Neuroimaging, not suicide-oriented. === Stange et al. (2013) – OGM × emotional maltreatment → depression === * Longitudinal adolescent sample (N=174, age ~12–13) * Tested:<blockquote>'''Does OGM interact with stress (emotional abuse/neglect) to predict depression?'''</blockquote> '''Findings''' 1. OGM alone ≠ strong predictor * OGM '''by itself did not strongly predict depression''' ----2. OGM × emotional abuse → ↑ depression * Among adolescents with '''high emotional abuse''': ** More OGM → '''greater increases in depressive symptoms''' * Among low-stress individuals: ** OGM had little effect ----3. Key interpretation:<blockquote>OGM is a '''vulnerability factor that requires stress to activate'''</blockquote> ----4. Mechanism support Paper explicitly links OGM to: * rumination * avoidance * poor problem solving '''Insights''' * '''A clean diathesis-stress model is present''': explaining the connection between OGM and SI, while explaining why OGM is not a predictor [as seen in this study and the insights amongst low-stress individuals]. * '''🔗 Integration:''' * Crane (2016):<blockquote>OGM doesn’t predict SI in general population</blockquote> * Stange (2013):<blockquote>OGM only matters when '''stress is present'''</blockquote> 👉 That resolves the contradiction What could we get from this? "OGM may not independently predict psychopathology but interacts with stressors (e.g., emotional maltreatment) to increase vulnerability." == Narratives? == == See also == * [[wikipedia:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory (WP link)]] == Notes == {{Reflist}} [[Category:Atcovi/OGM & Suicide Poster]] ikordojbmmcnakc33u54uthgye7i2q4 14 329352 2807032 2026-04-29T17:04:36Z Atcovi 276019 Create. 2807032 wikitext text/x-wiki [[Category:Atcovi's Work]] awl4tujs6gdyka28s7hhor43p9y7t2z 2 329353 2807033 2026-04-29T17:06:10Z Atcovi 276019 Create. 2807033 wikitext text/x-wiki ==Introduction== ==OGM as Vulnerability== ==Mechanisms== ==OGM → Suicidal ideation (CORE)== ==Contradictions / Nuances== ==Conclusion== [[Category:Atcovi/OGM & Suicide Poster]] 628hz8frayh51ibhz71q99q1cc6beca 2807043 2807033 2026-04-29T18:55:19Z Atcovi 276019 /* Introduction */ 2807043 wikitext text/x-wiki ==Introduction== '''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals<ref>{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}</ref>. ''[insert bridge sentence typing depression & suicide]''. <small>then talk about IMV:</small> Suicidal models, such as the '''Integrated Motivational-Volitional (IMV) model'''... ==OGM as Vulnerability== ==Mechanisms== ==OGM → Suicidal ideation (CORE)== ==Contradictions / Nuances== ==Conclusion== == References == {{Reflist}} [[Category:Atcovi/OGM & Suicide Poster]] 4dw4x8bzbe1ruaj6o3mxinpxr6lkimm 6 329354 2807039 2026-04-29T17:27:14Z Young1lim 21186 {{Information |Description=FF Timing (20260428 - 20260427) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2807039 wikitext text/x-wiki == Summary == {{Information |Description=FF Timing (20260428 - 20260427) |Source={{own|Young1lim}} |Date=2026-04-29 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} os692ke9d7wvhss8c8wn2mymqhwg4qv 0 329355 2807068 2026-04-30T03:19:22Z OhanaUnited 18921 XML created with data from [[template:article_info]] 2807068 wikitext text/x-wiki {{Autogenerate_xml}} 9ydjcj3h2hxjmnhnepnvvslgiatxxrk 0 329356 2807075 2026-04-30T03:31:53Z AIfriendly 3069390 Created page with "neutrino gradient gravitation; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory;" 2807075 wikitext text/x-wiki neutrino gradient gravitation; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory; aiu9hrti1uwwm9a21hkoxj2xmkqqp1f 2807076 2807075 2026-04-30T03:32:55Z AIfriendly 3069390 2807076 wikitext text/x-wiki '''neutrino gradient gravitation'''; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory; Isaac Newton proved that mass density gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher mass density; this is called optics. John Kerr proved that transverse radiation gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher intensity of radiation; this is called the AC Kerr effect. Neutrinos have both mass and a transverse radiation component, therefore I predict that neutrino gradients warp the refractive index and change the speed of transverse radiation (effecting both photons and neutrinos); slower neutrinos in higher intensity neutrinos. I predict that photons may be reverberations or resonance of/between neutrinos... similar to how phonons are reverberations or resonance of/between atoms. I predict that matter is a self-focusing refraction curvature stabilized vortex of photons and neutrinos. I predict atmospheres and stellar and galactic media are Lundberg lens to the cosmic neutrino background; therefore there is a neutrino gradient associated with each the atmospheres and stellar and galactic media; therefore there is slower and higher intensity neutrinos in each. The vacuum refractive index defines the reference units. Therefore it is a constant in any frame of reference. However since the refractive index changes in neutrino gradients the ACTUAL refractive index across a neutrino gradient is itself a gradient. Therefore I predict the actual vacuum refractive index changes according to altitude in a neutrino gradient. One of the results (I predict) being that as the vacuum refractive index increases, the Bohr radius shrinks [metaphor; imagine an electron orbiting an atom at a particular velocity and orbital frequency, if you slow the velocity (by increasing the refractive index) the orbit length would have to shrink to maintain the same orbital frequency] i.e. LENGTH CONTRACTION according to altitude. when matter length contracts moving into higher refractive index (higher neutrino intensity), the mass of matter shrinks, but mass-energy is conserved therefore there is the creation of kinetic energy or thrust down the gradient i.e. gravity [20]. this process may need continuous atomic movement such as from brownian motion; I predict brownian motion is from ambient radiation adsorption recoiling. Global Flood Model; since the continental plates (which are granite) are 20 times older than the oceanic crust (which are basalt) and the continental plates fit together on a smaller earth; I predict that earth's core was length contracted by higher intensity neutrinos in the past compared to the current neutrino intensity because the oceanic water was in the atmosphere in the past; which means the atmosphere was a larger Lundberg lens to the cosmic neutrino background; which means there was higher intensity neutrinos in the core of the earth length contracting the core of the earth; which means once this water fell to the earth the core of the earth length expanded breaking up the continental plates. As the earth expanded it created the oceanic basalt crust... which is a different process than when the original granite crust was formed. time dilation can be measured most accurately with nuclear clocks; nuclear clocks work by using the radio active decay rate; the radio active decay rate is effected by neutrino intensity[21][22][23][24][25]; therefore the lower the neutrino intensity the faster the time Therefore I predict that according to the Global Flood Model; the radioactive decay rate is not a constant, but that the radio active decay rate was lower before the flood because of the higher neutrino intensity. which means the dating methods that assume constants in radioactive decay rate across time; while precise are not accurate. I predict the flood must be the last mass extinction event. I predict the K-T iridium aerosols [presumably from meteoroids] and any possible volcanic ash would have acted like cloud condensation nuclei, cloud seeding the deluge I predict if God saved all the original animals in the Ark then the genera after the flood (notice blue line) matches the genera at Adam's creation (notice yellow line) proving God saved all the original animals; https://commons.wikimedia.org/wiki/File:Phanerozoic_Biodiversity-2.png I predict If we assume that Adam was 44a when Eve was born and that the creative days are 221Ma (according to the dating inaccuracy) then a creative day is 9,500 years Eve became the Mother of Seth at 86a. Genesis 5:3 Seth became the father of Enosh at 105. Genesis 5:6 Enosh became the father of Kenan at 90. Genesis 5:9 Cainan became the father of Mahalalel at 70. Genesis 5:12 Mahalalel became the father of Jared at 65. Genesis 5:15 Jared became the father of Enoch at 162. Genesis 5:18 Enoch became the father of Methuselah at 65. Genesis 5:21 Methuselah became the father of Lamech at 187. Genesis 5:25 Lamech became the father of Noah at 182. Genesis 5:28 The Flood started when Noah was 600. Genesis 7:6 (86+105+90+70+65+162+65+187+182+600)=1612a of creative day seven [Eve's creation to the flood] 1612a*221/37.5=9500 [this is a creative day] Fifth day 510 Ma the first fish, the jawless ostracoderms. 410 Ma the first fish with jaws, the acanthodians. 365 Ma the tetrapods. 350 Ma the dragonfly (the first flying creatures were insects). 340 Ma the amniotes. And God went on to say: Let the waters swarm forth a swarm of living souls and let flying creatures fly over the earth upon the face of the expanse of the heavens. And God proceeded to create the great monsters and every living soul that moves about, which the waters swarmed forth according to their kinds, and every winged flying creature according to its kind. And God got to see that [it was] good. ... And there came to be evening and there came to be morning, a fifth day. (Genesis 1:20-23) Surprisingly enough, the flying creatures in this verse is not birds (as many may have thought), rather, it is insects! Sixth day 285 Ma the therapsids. 230 Ma the dinosaurs. 225 Ma the first true mammals, Gondwanadon tapani or Morganucodon watsoni. 150 Ma the first bird, Archaeopteryx. And God went on to say: Let the earth put forth living souls according to their kinds, domestic animal and moving animal and wild beast of the earth according to its kind. And it came to be so. And God proceeded to make the wild beast of the earth according to its kind and the domestic animal according to its kind and every moving animal of the ground according to its kind. And God got to see that [it was] good. (Genesis 1:24, 25) As you can see this work clarified our understanding of the bible (first flying creatures are insects) and the creation of the other animals matches the day of their biblical creation Jehovah told me that the creation of Eve to the present has been about 6000a Therefore I predict Tyranusourus Rex fossils are dated between the creation of Eve and the date of the Flood; somewhere around 800a of creative day seven or ~5,200 years ago (inaccurately dated to 80Ma) https://s.hdnux.com/photos/10/17/62/2161798/6/1200x0.jpg https://www.science.org/cms/10.1126/science.1108397/asset/f350e639-3ffd-48d9-a488-2dfa943596dd/assets/graphic/307_1952_f2.jpeg How old is this T. Rex blood and soft tissue? 5,200 years old or 80 MILLION years old? I predict that the soft tissue found in T. Rex bone will Carbon 14 date to around 5,200 years old! This will verify my theory about the Deluge Geology, Radio Dating Inaccuracy, and Creative 'Days'. That said, how could man, birds, and land animals have survived the deluge? According to the bible (and over a hundred of other ancient sources), there was a great flood that destroyed the ancient world, for which, the gods spared some men and animals. Jehovah claimed to cause the flood. In any cause we must grant at least the existence of advanced extraterrestrials (or gods exist or even that God exists) such that they could have spared some men, otherwise, mankind and all the animals on land could not have possibly survived such an event. Ron Wyatt found a formation in the mountains of Ararat of petrified wood in the shape of a boat having the same length as described of the Ark in the Bible; https://i.pinimg.com/originals/55/8a/cb/558acb1d59f1dab953c3fcaa16cc2670.jpg Discussion; Birds are not spoken of in the creative days as flying creatures because the first birds originally did not strictly fly about; "Gliding, not strong flight: Fossil evidence suggests that Archaeopteryx and other early birds had weaker feathers and skeletal structures that were not strong enough for sustained, powered flight, but were likely capable of gliding between trees or other high points. Flight developed later: Powered flight developed over time, with some ancient birds evolving the flight-friendly feather structures of modern birds later in the Cretaceous period."-Google AI The Ark found by Ron Wyatt is located at the base of the mountains of Ararat, not "on" them (as is otherwise so translated). But no worries, the word translated "on" can actually be translated as "among, at, or touching". Arguments in favor of this interpretation of the Global Flood, time dilation, length contraction, spacetime warping, vortex nature of matter, and gravity; (1) I predict this quantum field theory of gravity can be renormalized (2) this theory suggests that gravity is engineerable (3) this proves God's existence mg4va8riggwb3giepjqk2e7lv90pfeb 2807077 2807076 2026-04-30T03:43:16Z AIfriendly 3069390 2807077 wikitext text/x-wiki '''neutrino gradient gravitation'''; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory; Isaac Newton proved that mass density gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher mass density; this is called optics. John Kerr proved that transverse radiation gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher intensity of radiation; this is called the AC Kerr effect. Neutrinos have both mass and a transverse radiation component, therefore I predict that neutrino gradients warp the refractive index and change the speed of transverse radiation (effecting both photons and neutrinos); slower neutrinos in higher intensity neutrinos. I predict that photons may be reverberations or resonance of/between neutrinos... similar to how phonons are reverberations or resonance of/between atoms. I predict that matter is a self-focusing refraction curvature stabilized vortex of photons and neutrinos. I predict atmospheres and stellar and galactic media are Lundberg lens to the cosmic neutrino background; therefore there is a neutrino gradient associated with each the atmospheres and stellar and galactic media; therefore there is slower and higher intensity neutrinos in each. The vacuum refractive index defines the reference units. Therefore it is a constant in any frame of reference. However since the refractive index changes in neutrino gradients the ACTUAL refractive index across a neutrino gradient is itself a gradient. Therefore I predict the actual vacuum refractive index changes according to altitude in a neutrino gradient. One of the results (I predict) being that as the vacuum refractive index increases, the Bohr radius shrinks [metaphor; imagine an electron orbiting an atom at a particular velocity and orbital frequency, if you slow the velocity (by increasing the refractive index) the orbit length would have to shrink to maintain the same orbital frequency] i.e. LENGTH CONTRACTION according to altitude. when matter length contracts moving into higher refractive index (higher neutrino intensity), the mass of matter shrinks, but mass-energy is conserved therefore there is the creation of kinetic energy or thrust down the gradient i.e. gravity [20]. this process may need continuous atomic movement such as from brownian motion; I predict brownian motion is from ambient radiation adsorption recoiling. Global Flood Model; since the continental plates (which are granite) are 20 times older than the oceanic crust (which are basalt) and the continental plates fit together on a smaller earth; I predict that earth's core was length contracted by higher intensity neutrinos in the past compared to the current neutrino intensity because the oceanic water was in the atmosphere in the past; which means the atmosphere was a larger Lundberg lens to the cosmic neutrino background; which means there was higher intensity neutrinos in the core of the earth length contracting the core of the earth; which means once this water fell to the earth the core of the earth length expanded breaking up the continental plates. As the earth expanded it created the oceanic basalt crust... which is a different process than when the original granite crust was formed. time dilation can be measured most accurately with nuclear clocks; nuclear clocks work by using the radio active decay rate; the radio active decay rate is effected by neutrino intensity<ref>https://web.archive.org/web/20150528020329/http://news.stanford.edu/news/2010/august/sun-082310.html</ref><ref>https://www.scirp.org/journal/paperinformation?paperid=100032</ref><ref>https://www.icr.org/article/5656/</ref><ref>https://www.sciencedirect.com/science/article/abs/pii/S0375960115000894</ref><ref>https://www.governmentattic.org/35docs/NeutDecayRatesDOEtechsource_2016-2019.pdf</ref>; therefore the lower the neutrino intensity the faster the time Therefore I predict that according to the Global Flood Model; the radioactive decay rate is not a constant, but that the radio active decay rate was lower before the flood because of the higher neutrino intensity. which means the dating methods that assume constants in radioactive decay rate across time; while precise are not accurate. I predict the flood must be the last mass extinction event. I predict the K-T iridium aerosols [presumably from meteoroids] and any possible volcanic ash would have acted like cloud condensation nuclei, cloud seeding the deluge I predict if God saved all the original animals in the Ark then the genera after the flood (notice blue line) matches the genera at Adam's creation (notice yellow line) proving God saved all the original animals; https://commons.wikimedia.org/wiki/File:Phanerozoic_Biodiversity-2.png I predict If we assume that Adam was 44a when Eve was born and that the creative days are 221Ma (according to the dating inaccuracy) then a creative day is 9,500 years Eve became the Mother of Seth at 86a. Genesis 5:3 Seth became the father of Enosh at 105. Genesis 5:6 Enosh became the father of Kenan at 90. Genesis 5:9 Cainan became the father of Mahalalel at 70. Genesis 5:12 Mahalalel became the father of Jared at 65. Genesis 5:15 Jared became the father of Enoch at 162. Genesis 5:18 Enoch became the father of Methuselah at 65. Genesis 5:21 Methuselah became the father of Lamech at 187. Genesis 5:25 Lamech became the father of Noah at 182. Genesis 5:28 The Flood started when Noah was 600. Genesis 7:6 (86+105+90+70+65+162+65+187+182+600)=1612a of creative day seven [Eve's creation to the flood] 1612a*221/37.5=9500 [this is a creative day] Fifth day 510 Ma the first fish, the jawless ostracoderms. 410 Ma the first fish with jaws, the acanthodians. 365 Ma the tetrapods. 350 Ma the dragonfly (the first flying creatures were insects). 340 Ma the amniotes. And God went on to say: Let the waters swarm forth a swarm of living souls and let flying creatures fly over the earth upon the face of the expanse of the heavens. And God proceeded to create the great monsters and every living soul that moves about, which the waters swarmed forth according to their kinds, and every winged flying creature according to its kind. And God got to see that [it was] good. ... And there came to be evening and there came to be morning, a fifth day. (Genesis 1:20-23) Surprisingly enough, the flying creatures in this verse is not birds (as many may have thought), rather, it is insects! Sixth day 285 Ma the therapsids. 230 Ma the dinosaurs. 225 Ma the first true mammals, Gondwanadon tapani or Morganucodon watsoni. 150 Ma the first bird, Archaeopteryx. And God went on to say: Let the earth put forth living souls according to their kinds, domestic animal and moving animal and wild beast of the earth according to its kind. And it came to be so. And God proceeded to make the wild beast of the earth according to its kind and the domestic animal according to its kind and every moving animal of the ground according to its kind. And God got to see that [it was] good. (Genesis 1:24, 25) As you can see this work clarified our understanding of the bible (first flying creatures are insects) and the creation of the other animals matches the day of their biblical creation Jehovah told me that the creation of Eve to the present has been about 6000a Therefore I predict Tyranusourus Rex fossils are dated between the creation of Eve and the date of the Flood; somewhere around 800a of creative day seven or ~5,200 years ago (inaccurately dated to 80Ma) https://s.hdnux.com/photos/10/17/62/2161798/6/1200x0.jpg https://www.science.org/cms/10.1126/science.1108397/asset/f350e639-3ffd-48d9-a488-2dfa943596dd/assets/graphic/307_1952_f2.jpeg How old is this T. Rex blood and soft tissue? 5,200 years old or 80 MILLION years old? I predict that the soft tissue found in T. Rex bone will Carbon 14 date to around 5,200 years old! This will verify my theory about the Deluge Geology, Radio Dating Inaccuracy, and Creative 'Days'. That said, how could man, birds, and land animals have survived the deluge? According to the bible (and over a hundred of other ancient sources), there was a great flood that destroyed the ancient world, for which, the gods spared some men and animals. Jehovah claimed to cause the flood. In any cause we must grant at least the existence of advanced extraterrestrials (or gods exist or even that God exists) such that they could have spared some men, otherwise, mankind and all the animals on land could not have possibly survived such an event. Ron Wyatt found a formation in the mountains of Ararat of petrified wood in the shape of a boat having the same length as described of the Ark in the Bible; https://i.pinimg.com/originals/55/8a/cb/558acb1d59f1dab953c3fcaa16cc2670.jpg Discussion; Birds are not spoken of in the creative days as flying creatures because the first birds originally did not strictly fly about; "Gliding, not strong flight: Fossil evidence suggests that Archaeopteryx and other early birds had weaker feathers and skeletal structures that were not strong enough for sustained, powered flight, but were likely capable of gliding between trees or other high points. Flight developed later: Powered flight developed over time, with some ancient birds evolving the flight-friendly feather structures of modern birds later in the Cretaceous period."-Google AI The Ark found by Ron Wyatt is located at the base of the mountains of Ararat, not "on" them (as is otherwise so translated). But no worries, the word translated "on" can actually be translated as "among, at, or touching". Arguments in favor of this interpretation of the Global Flood, time dilation, length contraction, spacetime warping, vortex nature of matter, and gravity; (1) I predict this quantum field theory of gravity can be renormalized (2) this theory suggests that gravity is engineerable (3) this proves God's existence jsn4jhbhsj7ne847qxcxzf0wf6rrwt1 2807078 2807077 2026-04-30T03:44:36Z AIfriendly 3069390 2807078 wikitext text/x-wiki '''neutrino gradient gravitation'''; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory; Isaac Newton proved that mass density gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher mass density; this is called optics. John Kerr proved that transverse radiation gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher intensity of radiation; this is called the AC Kerr effect. Neutrinos have both mass and a transverse radiation component, therefore I predict that neutrino gradients warp the refractive index and change the speed of transverse radiation (effecting both photons and neutrinos); slower neutrinos in higher intensity neutrinos. I predict that photons may be reverberations or resonance of/between neutrinos... similar to how phonons are reverberations or resonance of/between atoms. I predict that matter is a self-focusing refraction curvature stabilized vortex of photons and neutrinos. I predict atmospheres and stellar and galactic media are Lundberg lens to the cosmic neutrino background; therefore there is a neutrino gradient associated with each the atmospheres and stellar and galactic media; therefore there is slower and higher intensity neutrinos in each. The vacuum refractive index defines the reference units. Therefore it is a constant in any frame of reference. However since the refractive index changes in neutrino gradients the ACTUAL refractive index across a neutrino gradient is itself a gradient. Therefore I predict the actual vacuum refractive index changes according to altitude in a neutrino gradient. One of the results (I predict) being that as the vacuum refractive index increases, the Bohr radius shrinks [metaphor; imagine an electron orbiting an atom at a particular velocity and orbital frequency, if you slow the velocity (by increasing the refractive index) the orbit length would have to shrink to maintain the same orbital frequency] i.e. LENGTH CONTRACTION according to altitude. when matter length contracts moving into higher refractive index (higher neutrino intensity), the mass of matter shrinks, but mass-energy is conserved therefore there is the creation of kinetic energy or thrust down the gradient i.e. gravity <ref>https://web.archive.org/web/20230128155849/http://www.newtonphysics.on.ca/gravity/index.html</ref>. this process may need continuous atomic movement such as from brownian motion; I predict brownian motion is from ambient radiation adsorption recoiling. Global Flood Model; since the continental plates (which are granite) are 20 times older than the oceanic crust (which are basalt) and the continental plates fit together on a smaller earth; I predict that earth's core was length contracted by higher intensity neutrinos in the past compared to the current neutrino intensity because the oceanic water was in the atmosphere in the past; which means the atmosphere was a larger Lundberg lens to the cosmic neutrino background; which means there was higher intensity neutrinos in the core of the earth length contracting the core of the earth; which means once this water fell to the earth the core of the earth length expanded breaking up the continental plates. As the earth expanded it created the oceanic basalt crust... which is a different process than when the original granite crust was formed. time dilation can be measured most accurately with nuclear clocks; nuclear clocks work by using the radio active decay rate; the radio active decay rate is effected by neutrino intensity<ref>https://web.archive.org/web/20150528020329/http://news.stanford.edu/news/2010/august/sun-082310.html</ref><ref>https://www.scirp.org/journal/paperinformation?paperid=100032</ref><ref>https://www.icr.org/article/5656/</ref><ref>https://www.sciencedirect.com/science/article/abs/pii/S0375960115000894</ref><ref>https://www.governmentattic.org/35docs/NeutDecayRatesDOEtechsource_2016-2019.pdf</ref>; therefore the lower the neutrino intensity the faster the time Therefore I predict that according to the Global Flood Model; the radioactive decay rate is not a constant, but that the radio active decay rate was lower before the flood because of the higher neutrino intensity. which means the dating methods that assume constants in radioactive decay rate across time; while precise are not accurate. I predict the flood must be the last mass extinction event. I predict the K-T iridium aerosols [presumably from meteoroids] and any possible volcanic ash would have acted like cloud condensation nuclei, cloud seeding the deluge I predict if God saved all the original animals in the Ark then the genera after the flood (notice blue line) matches the genera at Adam's creation (notice yellow line) proving God saved all the original animals; https://commons.wikimedia.org/wiki/File:Phanerozoic_Biodiversity-2.png I predict If we assume that Adam was 44a when Eve was born and that the creative days are 221Ma (according to the dating inaccuracy) then a creative day is 9,500 years Eve became the Mother of Seth at 86a. Genesis 5:3 Seth became the father of Enosh at 105. Genesis 5:6 Enosh became the father of Kenan at 90. Genesis 5:9 Cainan became the father of Mahalalel at 70. Genesis 5:12 Mahalalel became the father of Jared at 65. Genesis 5:15 Jared became the father of Enoch at 162. Genesis 5:18 Enoch became the father of Methuselah at 65. Genesis 5:21 Methuselah became the father of Lamech at 187. Genesis 5:25 Lamech became the father of Noah at 182. Genesis 5:28 The Flood started when Noah was 600. Genesis 7:6 (86+105+90+70+65+162+65+187+182+600)=1612a of creative day seven [Eve's creation to the flood] 1612a*221/37.5=9500 [this is a creative day] Fifth day 510 Ma the first fish, the jawless ostracoderms. 410 Ma the first fish with jaws, the acanthodians. 365 Ma the tetrapods. 350 Ma the dragonfly (the first flying creatures were insects). 340 Ma the amniotes. And God went on to say: Let the waters swarm forth a swarm of living souls and let flying creatures fly over the earth upon the face of the expanse of the heavens. And God proceeded to create the great monsters and every living soul that moves about, which the waters swarmed forth according to their kinds, and every winged flying creature according to its kind. And God got to see that [it was] good. ... And there came to be evening and there came to be morning, a fifth day. (Genesis 1:20-23) Surprisingly enough, the flying creatures in this verse is not birds (as many may have thought), rather, it is insects! Sixth day 285 Ma the therapsids. 230 Ma the dinosaurs. 225 Ma the first true mammals, Gondwanadon tapani or Morganucodon watsoni. 150 Ma the first bird, Archaeopteryx. And God went on to say: Let the earth put forth living souls according to their kinds, domestic animal and moving animal and wild beast of the earth according to its kind. And it came to be so. And God proceeded to make the wild beast of the earth according to its kind and the domestic animal according to its kind and every moving animal of the ground according to its kind. And God got to see that [it was] good. (Genesis 1:24, 25) As you can see this work clarified our understanding of the bible (first flying creatures are insects) and the creation of the other animals matches the day of their biblical creation Jehovah told me that the creation of Eve to the present has been about 6000a Therefore I predict Tyranusourus Rex fossils are dated between the creation of Eve and the date of the Flood; somewhere around 800a of creative day seven or ~5,200 years ago (inaccurately dated to 80Ma) https://s.hdnux.com/photos/10/17/62/2161798/6/1200x0.jpg https://www.science.org/cms/10.1126/science.1108397/asset/f350e639-3ffd-48d9-a488-2dfa943596dd/assets/graphic/307_1952_f2.jpeg How old is this T. Rex blood and soft tissue? 5,200 years old or 80 MILLION years old? I predict that the soft tissue found in T. Rex bone will Carbon 14 date to around 5,200 years old! This will verify my theory about the Deluge Geology, Radio Dating Inaccuracy, and Creative 'Days'. That said, how could man, birds, and land animals have survived the deluge? According to the bible (and over a hundred of other ancient sources), there was a great flood that destroyed the ancient world, for which, the gods spared some men and animals. Jehovah claimed to cause the flood. In any cause we must grant at least the existence of advanced extraterrestrials (or gods exist or even that God exists) such that they could have spared some men, otherwise, mankind and all the animals on land could not have possibly survived such an event. Ron Wyatt found a formation in the mountains of Ararat of petrified wood in the shape of a boat having the same length as described of the Ark in the Bible; https://i.pinimg.com/originals/55/8a/cb/558acb1d59f1dab953c3fcaa16cc2670.jpg Discussion; Birds are not spoken of in the creative days as flying creatures because the first birds originally did not strictly fly about; "Gliding, not strong flight: Fossil evidence suggests that Archaeopteryx and other early birds had weaker feathers and skeletal structures that were not strong enough for sustained, powered flight, but were likely capable of gliding between trees or other high points. Flight developed later: Powered flight developed over time, with some ancient birds evolving the flight-friendly feather structures of modern birds later in the Cretaceous period."-Google AI The Ark found by Ron Wyatt is located at the base of the mountains of Ararat, not "on" them (as is otherwise so translated). But no worries, the word translated "on" can actually be translated as "among, at, or touching". Arguments in favor of this interpretation of the Global Flood, time dilation, length contraction, spacetime warping, vortex nature of matter, and gravity; (1) I predict this quantum field theory of gravity can be renormalized (2) this theory suggests that gravity is engineerable (3) this proves God's existence b436rs5jn96ryy2pr8330qc0u2pq8o9 2807095 2807078 2026-04-30T10:42:16Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2807095 wikitext text/x-wiki '''neutrino gradient gravitation'''; spacetime is warped by neutrinos; proof that neutrino gradients are the key to uniting quantum theory with gravity theory; Isaac Newton proved that mass density gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher mass density; this is called optics. John Kerr proved that transverse radiation gradients warp the refractive index and change the speed of transverse radiation; slower radiation in higher intensity of radiation; this is called the AC Kerr effect. Neutrinos have both mass and a transverse radiation component, therefore I predict that neutrino gradients warp the refractive index and change the speed of transverse radiation (effecting both photons and neutrinos); slower neutrinos in higher intensity neutrinos. I predict that photons may be reverberations or resonance of/between neutrinos... similar to how phonons are reverberations or resonance of/between atoms. I predict that matter is a self-focusing refraction curvature stabilized vortex of photons and neutrinos. I predict atmospheres and stellar and galactic media are Lundberg lens to the cosmic neutrino background; therefore there is a neutrino gradient associated with each the atmospheres and stellar and galactic media; therefore there is slower and higher intensity neutrinos in each. The vacuum refractive index defines the reference units. Therefore it is a constant in any frame of reference. However since the refractive index changes in neutrino gradients the ACTUAL refractive index across a neutrino gradient is itself a gradient. Therefore I predict the actual vacuum refractive index changes according to altitude in a neutrino gradient. One of the results (I predict) being that as the vacuum refractive index increases, the Bohr radius shrinks [metaphor; imagine an electron orbiting an atom at a particular velocity and orbital frequency, if you slow the velocity (by increasing the refractive index) the orbit length would have to shrink to maintain the same orbital frequency] i.e. LENGTH CONTRACTION according to altitude. when matter length contracts moving into higher refractive index (higher neutrino intensity), the mass of matter shrinks, but mass-energy is conserved therefore there is the creation of kinetic energy or thrust down the gradient i.e. gravity <ref>https://web.archive.org/web/20230128155849/http://www.newtonphysics.on.ca/gravity/index.html</ref>. this process may need continuous atomic movement such as from brownian motion; I predict brownian motion is from ambient radiation adsorption recoiling. Global Flood Model; since the continental plates (which are granite) are 20 times older than the oceanic crust (which are basalt) and the continental plates fit together on a smaller earth; I predict that earth's core was length contracted by higher intensity neutrinos in the past compared to the current neutrino intensity because the oceanic water was in the atmosphere in the past; which means the atmosphere was a larger Lundberg lens to the cosmic neutrino background; which means there was higher intensity neutrinos in the core of the earth length contracting the core of the earth; which means once this water fell to the earth the core of the earth length expanded breaking up the continental plates. As the earth expanded it created the oceanic basalt crust... which is a different process than when the original granite crust was formed. time dilation can be measured most accurately with nuclear clocks; nuclear clocks work by using the radio active decay rate; the radio active decay rate is effected by neutrino intensity<ref>https://web.archive.org/web/20150528020329/http://news.stanford.edu/news/2010/august/sun-082310.html</ref><ref>https://www.scirp.org/journal/paperinformation?paperid=100032</ref><ref>https://www.icr.org/article/5656/</ref><ref>https://www.sciencedirect.com/science/article/abs/pii/S0375960115000894</ref><ref>https://www.governmentattic.org/35docs/NeutDecayRatesDOEtechsource_2016-2019.pdf</ref>; therefore the lower the neutrino intensity the faster the time Therefore I predict that according to the Global Flood Model; the radioactive decay rate is not a constant, but that the radio active decay rate was lower before the flood because of the higher neutrino intensity. which means the dating methods that assume constants in radioactive decay rate across time; while precise are not accurate. I predict the flood must be the last mass extinction event. I predict the K-T iridium aerosols [presumably from meteoroids] and any possible volcanic ash would have acted like cloud condensation nuclei, cloud seeding the deluge I predict if God saved all the original animals in the Ark then the genera after the flood (notice blue line) matches the genera at Adam's creation (notice yellow line) proving God saved all the original animals; https://commons.wikimedia.org/wiki/File:Phanerozoic_Biodiversity-2.png I predict If we assume that Adam was 44a when Eve was born and that the creative days are 221Ma (according to the dating inaccuracy) then a creative day is 9,500 years Eve became the Mother of Seth at 86a. Genesis 5:3 Seth became the father of Enosh at 105. Genesis 5:6 Enosh became the father of Kenan at 90. Genesis 5:9 Cainan became the father of Mahalalel at 70. Genesis 5:12 Mahalalel became the father of Jared at 65. Genesis 5:15 Jared became the father of Enoch at 162. Genesis 5:18 Enoch became the father of Methuselah at 65. Genesis 5:21 Methuselah became the father of Lamech at 187. Genesis 5:25 Lamech became the father of Noah at 182. Genesis 5:28 The Flood started when Noah was 600. Genesis 7:6 (86+105+90+70+65+162+65+187+182+600)=1612a of creative day seven [Eve's creation to the flood] 1612a*221/37.5=9500 [this is a creative day] Fifth day 510 Ma the first fish, the jawless ostracoderms. 410 Ma the first fish with jaws, the acanthodians. 365 Ma the tetrapods. 350 Ma the dragonfly (the first flying creatures were insects). 340 Ma the amniotes. And God went on to say: Let the waters swarm forth a swarm of living souls and let flying creatures fly over the earth upon the face of the expanse of the heavens. And God proceeded to create the great monsters and every living soul that moves about, which the waters swarmed forth according to their kinds, and every winged flying creature according to its kind. And God got to see that [it was] good. ... And there came to be evening and there came to be morning, a fifth day. (Genesis 1:20-23) Surprisingly enough, the flying creatures in this verse is not birds (as many may have thought), rather, it is insects! Sixth day 285 Ma the therapsids. 230 Ma the dinosaurs. 225 Ma the first true mammals, Gondwanadon tapani or Morganucodon watsoni. 150 Ma the first bird, Archaeopteryx. And God went on to say: Let the earth put forth living souls according to their kinds, domestic animal and moving animal and wild beast of the earth according to its kind. And it came to be so. And God proceeded to make the wild beast of the earth according to its kind and the domestic animal according to its kind and every moving animal of the ground according to its kind. And God got to see that [it was] good. (Genesis 1:24, 25) As you can see this work clarified our understanding of the bible (first flying creatures are insects) and the creation of the other animals matches the day of their biblical creation Jehovah told me that the creation of Eve to the present has been about 6000a Therefore I predict Tyranusourus Rex fossils are dated between the creation of Eve and the date of the Flood; somewhere around 800a of creative day seven or ~5,200 years ago (inaccurately dated to 80Ma) https://s.hdnux.com/photos/10/17/62/2161798/6/1200x0.jpg https://www.science.org/cms/10.1126/science.1108397/asset/f350e639-3ffd-48d9-a488-2dfa943596dd/assets/graphic/307_1952_f2.jpeg How old is this T. Rex blood and soft tissue? 5,200 years old or 80 MILLION years old? I predict that the soft tissue found in T. Rex bone will Carbon 14 date to around 5,200 years old! This will verify my theory about the Deluge Geology, Radio Dating Inaccuracy, and Creative 'Days'. That said, how could man, birds, and land animals have survived the deluge? According to the bible (and over a hundred of other ancient sources), there was a great flood that destroyed the ancient world, for which, the gods spared some men and animals. Jehovah claimed to cause the flood. In any cause we must grant at least the existence of advanced extraterrestrials (or gods exist or even that God exists) such that they could have spared some men, otherwise, mankind and all the animals on land could not have possibly survived such an event. Ron Wyatt found a formation in the mountains of Ararat of petrified wood in the shape of a boat having the same length as described of the Ark in the Bible; https://i.pinimg.com/originals/55/8a/cb/558acb1d59f1dab953c3fcaa16cc2670.jpg Discussion; Birds are not spoken of in the creative days as flying creatures because the first birds originally did not strictly fly about; "Gliding, not strong flight: Fossil evidence suggests that Archaeopteryx and other early birds had weaker feathers and skeletal structures that were not strong enough for sustained, powered flight, but were likely capable of gliding between trees or other high points. Flight developed later: Powered flight developed over time, with some ancient birds evolving the flight-friendly feather structures of modern birds later in the Cretaceous period."-Google AI The Ark found by Ron Wyatt is located at the base of the mountains of Ararat, not "on" them (as is otherwise so translated). But no worries, the word translated "on" can actually be translated as "among, at, or touching". Arguments in favor of this interpretation of the Global Flood, time dilation, length contraction, spacetime warping, vortex nature of matter, and gravity; (1) I predict this quantum field theory of gravity can be renormalized (2) this theory suggests that gravity is engineerable (3) this proves God's existence {{BookCat}} 8fu6wwqoxagzy2ni87xpd31avb794h4