Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.7 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 0 2651 2815934 586944 2026-06-16T13:30:36Z Atcovi 276019 project box(es) 2815934 wikitext text/x-wiki {{chemistry}} {{collection}} ==Wikipedia== [[w:Chemistry]] [[w:Category:Chemistry]] ===Analytical chemistry=== * [[w: Cathodic stripping voltammetry]] ==Wikinews== [[n:Category:Science and technology topics]] ==Wikibooks== *[[b:A-level Chemistry|A-level Chemistry]] *[[b:Alchemy|Alchemy]] *[[b:Biochemistry|Biochemistry]] *[[b:Chemical synthesis|Chemical synthesis]] *[[b:Computational chemistry|Computational chemistry]] *[[b:Crystallography|Crystallography]] *[[b:General Chemistry|General Chemistry]] *[[b:IB Chemistry|IB Chemistry]] *[[b:Inorganic Chemistry|Inorganic Chemistry]] *[[b:Nanowiki - The opensource handbook of nanoscience|Nanowiki - The opensource handbook of nanoscience]] *[[b:Organic Chemistry|Organic Chemistry]] *[[b:Physical Chemistry|Physical Chemistry]] *[[b:Proteomics|Proteomics]] *[[b:VCE Chemistry|VCE Chemistry]] ==Commons== [[commons:Chemistry|Chemistry ]] [[Category:Chemistry]] nbsukqvjn30al0a7o6kettlkyxumf3f 102 4704 2815925 2812728 2026-06-16T12:49:10Z Contagious Owl 2872581 /* Western music theory */ 2815925 wikitext text/x-wiki [[File:Musical notes.svg|right|372x372px]] Music is a self-expressed art form that organizes sound and silence through rhythm, tempo, pitch, melody, harmony, timbre, texture, dynamics, and other acoustic or electronic means, including instruments, synthesis, sampling, and environmental sound. Whether a work is recognized as music often depends on cultural context and the shared understanding of performers and listeners. Music may be created individually or collectively through ensembles, orchestras, groups, and other forms of participation. While atonal music exists, many musical traditions are characterized by melody and harmony. Rhythm encompasses tempo, meter, and articulation, while dynamics describe variations in loudness. Timbre and texture refer to the distinctive qualities and combinations of sounds, sometimes described as the "color" of music. ==Music and aural theory== ===[[Fundamentals of Music|Fundamentals of music]]=== Music is organized sound, combining pitch, rhythm, timbre, dynamics, texture, and form. These elements provide us a common vocabulary for describing how music is structured and experienced. Fundamentals of Music serves as a foundation for later study in theory, performance, composition, and musicology, introducing the basic concepts used to analyze and discuss music regardless of genre or cultural origin. * [[Introduction to music]] (To be absorbed into Fundamentals of music) ===Western music theory=== The goal of this section is to equip the student with the tools and skills necessary to compose, arrange and analyze music. Upon studies of these resources, students will possess the skills and knowledge of western theory, creative writing, arranging, as well as having a portfolio of original works. {{colbegin|3}}* [[Music Theory I: Basics]] ** [[Music Notation]] ** [[Scales (music)]] ** [[Key Signatures]] ** [[Meter and Time Signatures]] * [[Harmony]] * [[Music Theory II: Harmony 1]] ** [[Triads]] ** [[Cadence]] ** [[Inversions (music)]] * [[Chords (music)|Chords]] * [[Music Theory III: Harmony 2]] ** [[Seventh chords]] ** [[Modulation]] ** [[Modal mixture]] ** [[Chromatic Harmony]] * [[Music Theory IV: Advanced Analysis]] ** [[Form and Analysis|Form and analysis]] ** [[Counterpoint]] ** [[Fugue]] * [[Music Theory V: Developing Frontiers]] ** [[Non-functional Harmony]] ** [[Xenharmonic Music Theory]] ** [[Xenrhythm]] * [[Music Appreciation|Music appreciation]] * [[w:Glossary of music terminology|Glossary of music terminology]]{{colend}} === Non-western music theory=== Some Non-European cultures have different music composition, arrangement and analysis traditions, less commonly known in western cultural spheres.{{colbegin|3}} * [[Gamelan music theory]] * [[Carnatic music theory]] * [[Andalusi classical music theory]] * [[Persian classical music theory]] * [[Arabic classical music theory]] * [[Ottoman classical music theory]] * [[Hindustani classical music theory]]{{colend}} ===Aural theory and ear training=== Ear training is learning/training your ears to recognize what you hear and put it down onto paper. These are basic learning guides, exercises and projects to help you understand in a meaningful way the flurry of sound in music.{{colbegin|3}} * [[Ear training - Intervals and Harmony|Ear training - Intervals and Harmony (pitch oriented)]] * [[Ear training - rhythm]] * [[Sight singing]] * [[Transcription (music)]] {{colend}} ===Genres=== Some genres of Western music have genre-specific music theory.{{colbegin|3}} * [[Basic Blues & Rock]] * [[Country music]] * [[Folk music]] * [[Jazz]]{{colend}} ==Composition== {{colbegin|3}} * [[Beginning composition]] * [[Advanced composition]] * [[Lyrical composition|Lyrical composition]] * [[Arranging]] * [[Orchestration]] * [[Film scoring for Musicians|Film scoring]] (in conjunction with the [[Course:Practical narrative film editing|Film editing course]]) * [[Final Theory Project]] ===[[Music Technology]]=== *[[DAWs]] * [[MIDI]] {{colend}} ==Performance== {{colbegin|3}} * [[Solo performance]] * [[Ensemble performance]] * [[Improvisation (music)]] * [[Conducting]] * [[Music pedagogy]] {{colend}} ==Musicology== {{colbegin|3}} ===General Musicology=== * [[Music Appreciation|Music appreciation and history]] * [[Survey of Musical Genres|Survey of musical genres]] * [[Music in Film|Music in film]] * [[The Symphony and the Opera|Symphony and opera]] ===Historical Musicology=== ====Western Music History==== * [[Brief History of Western Music]] * [[Music of the Medieval Era]] * [[Music of the Renaissance]] * [[Music of the Baroque Era]] * [[Music of the Classical Era]] * [[Music of the Romantic Era]] * [[Music of the 20th Century]] ====Nonwestern Music History==== * [[Gamelan music history]] * [[Carnatic music history]] * [[Andalusi classical music history]] * [[Persian classical music history]] * [[Arabic classical music history]] * [[Ottoman classical music history]] * [[Hindustani classical music ]] ===Ethnomusicology=== * [[Folk Music]] * [[Indigenous Music]] * [[Comparative Ethnomusicology]] {{colend}} ==Music instruments== {{MultiCol}} === [[String instruments]] === * [[Violin]] * [[Viola]] * [[Violoncello]] * [[Double bass]] * [[Fiddle]] * [[Harp]] * [[Guitar]] ** [[Classical guitar|Classic guitar (or ''"Acoustic"'' guitar)]] ** [[Electric Guitar|Electric guitar]] ** [[Bass guitar|Bass guitar]] * [[Ukulele]] * [[Banjo]] * [[Mandolin]] * [[Lute]] {{ColBreak}} === [[Woodwind instruments]] === * [[Flute]] * [[Oboe]] * [[Clarinet]] * [[Bassoon]] * [[Saxophone]] ** [[Soprano Saxophone|Soprano saxophone]] ** [[Alto Saxophone|Alto saxophone]] ** [[Tenor Saxophone|Tenor saxophone]] ** [[Baritone Saxophone|Baritone saxophone]] * [[Recorder]] * [[Ocarina]] ===[[Brass instruments]]=== * [[Trumpet]] * [[French horn]] * [[Trombone]] * [[Tuba]] * [[Euphonium]] {{ColBreak}} === [[Percussion instruments]] === * [[Concert Percussion|Concert percussion]] (Snare drum, crash cymbals, timpani, etc.) * [[Drum set]] * [[Mallet Instruments|Mallet instruments]] (Marimba, xylophone, vibraphone, chimes, etc.) * [[Tabla]] (an Indian pair of drums) * [[Pipe and tabor]] === [[Keyboard instruments]] === * [[Piano]] * [[Organ]] === [[Topic: Voice | Voice]] === * [[Soprano]] * [[Contralto]] * [[Countertenor]] * [[Tenor]] * [[Baritone]] * [[Bass (voice)|Bass]] {{EndMultiCol}} ==Music resources== [[wikibooks:Subject:Music|Wikibooks - Music]]{{MultiCol}} === Hands on === * [[Blues basics]] * [[Rock basics]] * [[Wikiversity the Movie/music|Wikiversity the movie : music]] * [[Jamming Online|Jamming online]] * [[Experimental music]] * [[Film scoring for Musicians|Practical lessons in film scoring]] (in conjunction with the [[Course:Practical narrative film editing|film editing course]]) * [[Digital Audio Workstation]] === Textbooks === * [[b:Music|Wikibooks Music theory]] * [[b:Western Music History|Western music history]] * [[b:Sound Recording|Sound recording]] {{ColBreak}} === Open-Source software === ;For all operating systems * [http://openmetronome.sourceforge.net/ Metronome] * [http://sourceforge.net/projects/vtone/ Vtones] (Basic Midi editor) * [http://audacity.sourceforge.net/ Audacity] (Sound editor) * [http://ardour.org/ Ardour] (Digital Audio Workstation; A great program for multi-track recording, mixing, mastering, etc.) * [http://musescore.org/ Musescore] (Music Notation software) * [https://otuner.sourceforge.net/ Tuning Software] * [https://supercollider.github.io/ SuperCollider] (Programming Language and Environment for sound synthesis and algorithmic composition) {{ColBreak}} ;For Linux * [https://github.com/calf-studio-gear/calf/ Calf Plugins] Sound Plugins including compressor, multichorus, reverb,etc. * [http://www.antcom.de/gtick/ GTick] (very nice and useful metronome for Gnome desktop) * [[:w:LMMS|Linux MultiMedia Studio (''LMMS'')]] === External links === * [http://music.wikia.com/wiki/Music_Hub Music topics on Wikia] * [[w:Wikipedia:Sound/list|Musical works available for download]] * [[w:History of music|'History of music' on Wikipedia]] {{EndMultiCol}} ==Active participants== ''If you are an active participant in this school, you can list your name below. (this can help small schools grow and the participants communicate better)'' Please leave a timestamp - if it is more than a year old, there is potential for nomination to the inactive participants list. *[[User:Kirby_-_Electrotechnics|Kirby]] (he/him), Banjo, May 2026 ==Inactive participants== *[[User:CQ|CQ]] * Since 20 February 2012‎. Reviewed [[Portal:Pentatonic Impressionism (China Wu Sheng) in the view of Neo-classical Piano Techniques-training]] for Main Page News about 8 August 2019. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 19:58, 16 January 2020 (UTC) * [[User:SelfieCity|SelfieCity]] 12 July 2021 *[[User:HappyCamper|HappyCamper]] *[[User:Thierry613|Thierry613]] *[[User:Bibeyjj|Bibeyjj]] glzba9hmmjsk27xewtr099buoeypg8z 10 6746 2815930 2815914 2026-06-16T13:28:04Z Atcovi 276019 rv 2815930 wikitext text/x-wiki <onlyinclude> {{mbox | type = notice | image = [[Image:Historical.png|80px|alt=|link=]] | text = '''This Wikiversity page is inactive and is of [[Wikiversity:History of Wikiversity|historical interest]].''' You may discuss this page on the [[{{NAMESPACE}} talk:{{PAGENAME}}|talk page]], copy all or part of this page to another page for further editing or start a new discussion about this page at the [[Wikiversity:Colloquium]]. }} <includeonly>[[Category:Wikiversity archive]]</includeonly> </onlyinclude> <noinclude> This template adds pages to [[:Category:Wikiversity archive]]. [[Category:Page flag templates]] [[Category:Administrative templates]] </noinclude> exoslv9fgbvu7a9tj9k65arml13v3of 2815931 2815930 2026-06-16T13:28:17Z Atcovi 276019 Protected "[[Template:Historical]]": Highly visible template ([Edit=Allow only curators and custodians] (indefinite) [Move=Allow only curators and custodians] (indefinite)) 2815930 wikitext text/x-wiki <onlyinclude> {{mbox | type = notice | image = [[Image:Historical.png|80px|alt=|link=]] | text = '''This Wikiversity page is inactive and is of [[Wikiversity:History of Wikiversity|historical interest]].''' You may discuss this page on the [[{{NAMESPACE}} talk:{{PAGENAME}}|talk page]], copy all or part of this page to another page for further editing or start a new discussion about this page at the [[Wikiversity:Colloquium]]. }} <includeonly>[[Category:Wikiversity archive]]</includeonly> </onlyinclude> <noinclude> This template adds pages to [[:Category:Wikiversity archive]]. [[Category:Page flag templates]] [[Category:Administrative templates]] </noinclude> exoslv9fgbvu7a9tj9k65arml13v3of 2815932 2815931 2026-06-16T13:28:44Z Atcovi 276019 test 2815932 wikitext text/x-wiki <onlyinclude> {{ambox | type = style | image = [[Image:Historical.png|80px|alt=|link=]] | text = '''This Wikiversity page is inactive and is of [[Wikiversity:History of Wikiversity|historical interest]].''' 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Should you wish to revive discussion regarding the subject, you may try using the [[{{NAMESPACE}} talk:{{PAGENAME}}|talk page]] or start a new discussion at the [[Wikiversity:Colloquium]]. {{#if:{{{1|}}} |<td style="width: 1em"><th id="shortcut" class="noprint" style="border: 1px solid #999; background: #fff; margin: 0 0 .5em 1em; text-align: center; padding: 5px"><small><br />{{{1|}}}</small></th></td>}} |}<includeonly>[[Category:Wikiversity archive]]</includeonly><noinclude>This template adds pages to [[:Category:Wikiversity archive]]. [[Category:Templates|Historical]]</noinclude> <noinclude>[[Category:Page flag templates]]</noinclude> 0l9jn7e3ovgq10uzw9weys6i24gs105 3 54428 2815956 2809474 2026-06-16T14:46:11Z Regliste 3029369 /* Found a potential reviewer */ Reply 2815956 wikitext text/x-wiki Thank you for making this happen: [[User:OhanaUnited/Sister Projects Interview]] - I am sure your readers will profit from the better info from all here. Below more info about Wikiversity, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) ==Welcome== '''Hello OhanaUnited, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[Image:Signature_icon.png]] in the edit window makes it simple. To [[Wikiversity:Introduction|get started]], you may <div style="width:50.0%; float:left"> * [[Wikiversity:Guided tour|Take a guided tour]] and learn [[Help:Editing|to edit]]; * Explore our [[Portal:Learning Projects|learning projects]]; * [[Wikiversity:Browse|Browse]] our [[Wikiversity:Portals|portals]], [[Wikiversity:Schools|schools]], and [[Wikiversity:Research|research]] activities; </div> <div style="width:50.0%; float:left"> * Read and help develop our community [[Wikiversity:Policies|policies]]；or * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. </div> <br clear="both"/> And don't forget to [[Wikiversity:Introduction explore|explore]] Wikiversity with the links to your left. [[Wikiversity:Be bold|Be bold]], and see you around Wikiversity! ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) == Environmental experts needed :) == Hi OhanaUnited, There have been a number of environmental projects started here and there... a few I can think of offhand: *[[Project proposal:global warming]] -- I'm not sure where that stands now... it was one of the first proposals back in 2006 I think *[[Bloom Clock]] -- Essentially a phenology project... among other things the data collections will hopefully be handy for later projects tracking changes in bloom time as local and global temperature trends change *[[Radio Discussion/Living on Earth]] -- Something a couple of us were experimenting with this past winter, using a radio show as our "lecture" and collecting materials for further learning. I'm not by any means an expert in environmental science, but as a horticulurist and farmer I'm well-versed in managing my local ecology... let me know if you start something! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 15:06, 28 March 2008 (UTC) :See also [[:Category:Ecology]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 11:18, 29 March 2008 (UTC) == Commons == Is there a page on commons somewhere with the questions? I'm sure I could round up a few interested commonists on IRC if you give me a link :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:15, 30 March 2008 (UTC) == Clarifications == Hi OhanaUnited, I've asked some questions at [[User talk:OhanaUnited/Sister Projects Interview#Voice(s)]] - I'd appreciate if you could clarify before I contribute to your initiative. Thanks, [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:54, 1 April 2008 (UTC) == removing == I removed the signatures after names in order to move forward summarizing the answers... and then I saw that you said to not do that... I reverted... How would be best to summarize the answers? --[[User:Remi|Remi]] 04:05, 21 April 2008 (UTC) :I voiced a related question in the "Voice(s)" section on the talk page.. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:40, 21 April 2008 (UTC) == Publication date == Hi OhanaUnited, would you be able to let us know when [[User:OhanaUnited/Sister_Projects_Interview|your interview]] will be published? Perhaps either on the talk page or on the [[Wikiversity:Colloquium#User:OhanaUnited/Sister Projects Interview - the earliest publication date is April 21|Colloquium]]. Thanks. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:39, 21 April 2008 (UTC) == Font Tag == The font tag is now obsolete. Please adjust your signature to something like: <blockquote> <pre> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] </pre> </blockquote> Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:37, 29 May 2018 (UTC) == Reorganised discussion == This is to let you know that the discussion at [[Talk:WikiJournal User Group#Code of Conduct]] has been reorganised to ease constructive inputs that help in updating the [[WikiJournal User Group/Code of conduct draft|document]]. If you would like to summarily oppose implementation of any Code of Conduct, feel free to place your opposition at [[Talk:WikiJournal User Group#Discussion: Whether any Code of Conduct needs to be defined and implemented]]. For any other constructive inputs please feel free to do so at [[Talk:WikiJournal User Group#Discussion: Salient updates that need to be made to the existing draft]]. Thanks for your cooperation. <span style="font-family:Segoe script">[[w:User:Diptanshu Das|<b style="color:#f00">D</b><b style="color:#f60">ip</b><b style="color:#090">ta</b><b style="color:#00f">ns</b><b style="color:#60c">hu</b>]] [[User talk:Diptanshu Das|&#128172;]]</span> 12:20, 16 December 2018 (UTC) == Maps via Wikidata == I remember you were testing maybe plotting a map of editor locations. I've been testing [https://w.wiki/CGk generating a map in Wikidata]. If we include all journal editors on the WikiJournal's page then it's possible to find the geocoordinates of their employer. Eventually it should be automate-able via [[wikidata:Wikidata:Bot_requests#Automated_addition_of_WikiJournal_metadata_to_Wikidata|this bot request]], but would have to be done manually for now. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:22, 18 November 2019 (UTC) :Note, [https://w.wiki/CWP updated version] with better interface for multiple points. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 23 November 2019 (UTC) == Query at review page == I just noticed there's a query for you at [[Talk:WikiJournal Preprints/Working with Bipolar Disorder During the COVID-19 Pandemic: Both Crisis and Opportunity|this page]] (the editor forgot to ping, or is unaware of the practice). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:43, 17 May 2020 (UTC) == Re: A Phonological Analysis of Selected Nigerian Newscasters Rendition == I appreciate your consideration of my article for publication. However, you have not provided an email address where I could send the word version or preferably, I would like to be guided on how to get the article uploaded on wiki commons. Thank you. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 07:35, 25 August 2022 (UTC) == The Validity of [[WikiJournal Preprints/The Effect of Corticosteroids on the Mortality Rate in COVID-19 Patients, v2]] == Hello Andrew, I'm coming to you to ask whether the mentioned paper's topic/objective is suitable for publication in the WikiJournal of Medicine. I was going to extensively work on it this summer, but I wanted to get written confirmation that this paper would be suited for my time in developing it. I also wanted to see if a Wikijournal of Humanities paper on Meditation would be suitable. I'm not sure if you're familiar with that wikijournal's guidelines, but I figured it was worth asking. Thank you, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:20, 27 August 2022 (UTC) == Request == Please, I do not know whether you could help upload the article if I send its soft copy as MS word document or pdf to you. Thanks. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 03:44, 5 September 2022 (UTC) == Volunteering to help with WikiJournal of Humanities == I kinf of forgot about WikiJournals for a few years, and I am amazed at the progress made. Well, as a real-life professor of sociology, I'd be happy to help with WikiJournal of Humanities which seems to be closed to my field. Do let me know how I can help, assuming of course you need any assistance. (If you reply here, please ping me back, TIA). [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:31, 8 November 2022 (UTC) == In other news == I am a strict believer in learning from the bottoms up (as a teacher who tells students to edit Wikipedia, for example, I never ask them to do things I haven't done myself before). And it so happens, I have a publication that I think is within the scope of WikiJournal Medicine, and now that I know it is indexed in SCOPUS, it meets my university's requirements too. As I am not yet on the board or such, I think I have no COI, so I decided to went ahead and submit my work at [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia]] . Before I finish copyediting it (I think I need to upload images to Wikimedia Commons and reformat references to footnotes) and finish the rest of the submission procedure, can I ask you to confirm that this topic is within the scope of WJMED and our previous conversation does not create any COI for me to submit it (I am fine putting my editorial application fpr WJHUM from yesterday on hold for the duration of the review process, if necessary)? Oh, to confirm, WikiJournals allows and prefers non-anonymous submissions, right? So I don't need to anonymize citations to my own work, etc.? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:08, 10 November 2022 (UTC) :{{re|Piotrus}} Each WikiJournal (Medicine, Science, Humanities) has separate editorial boards, similar to how "Nature Medicine" and "Nature Chemistry" are two different journals, have different editor-in-chief and different ISSN/DOI even though they are both owned and published by Springer Nature. Each WikiJournal operates and makes article decisions independently from each other while sharing same pool of resources (hired contractors, H/R, overhead cost). Therefore, whether or not you are on the Humanities board will not cause a COI when submitting to Medicine. I am the managing editor for Science, so our conversations won't cause any COI. I will defer your question on whether your preprint falls into the scope of Medicine to [[User:Rwatson1955]], who is the managing editor for the Medicine journal. And yes, we [[WikiJournal of Medicine/Publishing#Duties_of_authors|ask that "authors should be given by real names in their articles"]] so there is no need to anonymize. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:58, 10 November 2022 (UTC) ::I submitted [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|my article]] two days ago and filled in a Google Form, which suggested I'd receive confirmation email, but nothing happened and the article still has a notice that it is not submitted for review. Any chance you could check from your end if things are fine or ping someone who can, as maybe I haven't clicked something correctly or such? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:09, 16 November 2022 (UTC) :::{{re|Piotrus}} That's my fault. Been busy with work. I'll process the new submissions today and update the status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:14, 16 November 2022 (UTC) == Concerning an article == Hello, I'm not sure if you are aware that I have written a new article on Wikiversity, entitled: [[WikiJournal Preprints/Orhan Gazi, the first statesman|Orhan Gazi, the first Statesman]], I started it in September 2022 and finished it in March of the same year, and I was hoping that finding some peer reviewers wouldn't take much time. However, the article remained as it was for more than a year, and I had to ask two professors I know personally to check my work, which they did and their notes were sent in pdf format and added [[Talk:WikiJournal Preprints/Orhan Gazi, the first statesman|here]]. Now the article still needs an editor, before it can be finalized and published, and a fellow Wikipedian, [[User:علاء|Alaa]], suggested your name. I hope that perhaps you could check it. Please let me know what you think, best wishes-- [[User:باسم|باسم]] ([[User talk:باسم|discuss]] • [[Special:Contributions/باسم|contribs]]) 20:17, 7 May 2023 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:WikiJournal Bioclogging - ES.pdf]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 15:41, 19 December 2023 (UTC) ==Japanese rendering== Thanks to your help, I could make [[WikiJournal_of_Science/Bioclogging/ja|Japanese translation of bioclogging article]]. I feel that display style of Japanese sentense is wierd, because breakline is restricted to some characters such as "、". Japanese does not break words with spaces, as normal in western languages, and therefore we break lines anywhere. For example, see [[w:ja:バイオクロッギング|Japanese edition of bioclogging article in Wikipedia]]. It can be fixed by using css. For example, in this paragraph バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。 We can set word-break: break-all, and then <span style="word-break: break-all">バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。</span> Setting this to all paragraphs may be a solution. I would like to know if there is a smarter way to do the same thing. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 08:55, 16 February 2024 (UTC) :@[[User:Katsutoshi Seki|Katsutoshi Seki]] Thanks for raising this issue. I can read and write in Chinese (and therefore I can read Japanese Kanji) so I understand what you're describing about the software not finding spaces to break up words to the next line. I have [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBioclogging%2Fja&diff=2606145&oldid=2605982 forced] the software to consider appropriate line break locations. I'm confident with the line breaks in Kanji but less so in Katakana and Hiragana. And I don't know how it may look like under different computer screens (or mobile phone). Please review and see if the line breaks are done accurately. Also, can you please provide a Japanese translation for the phrases "For the English translation, please see this link." and "For the Japanese translation, please see this link."? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:49, 16 February 2024 (UTC) :: Unfortunately, giving <nowiki>{{wbr}}</nowiki> to some places does not help much, because appropriate place for breaking line changes to various width of windows. Therefore, using <nowiki><span style="word-break: break-all"></nowiki> to all paragraphs, as I showed above, is necessary. I would like to know if there is an appropriate way to change the stylesheet in the page at once. For the translation, "For the English translation, please see '''this link'''."　to "英語版は'''このリンク'''参照", and "For the Japanese translation, please see '''this link'''." to "日本語版は'''このリンク'''参照" [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 01:39, 17 February 2024 (UTC) :::Thanks for verifying. I have removed {{tl|wbr}} and added <nowiki><span style="word-break: break-all"></nowiki>. It doesn't seem very effective to bulleted items. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 17 February 2024 (UTC) :::: I also added css to bulleted items. Now it works find. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 04:50, 17 February 2024 (UTC) :::: I created [[Template:BreakAll]] and applied. ChatGPT was helpful for creating the LUA module. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 12:55, 17 February 2024 (UTC) == Article progress == Hi Ohana, it was great to meet you at the conference in November. I finally got around to finishing the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As we discussed, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. Sorry for the very long delay on this article and I apologize if this is not the right forum to report progress. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 03:45, 21 February 2024 (UTC) :Hi @[[User:Buidhe|Buidhe]], our apologies for the very long delay in replying to you. [[User:Fransplace|Fransplace]], the editor-in-chief for WikiJournal of Humanities, will be looking at your submission shortly. Since we already received two reviewers' comments and you have completed your revisions, are you ok with continuing with the submission process? I think we are on the home stretch with very few items remaining. Can you add your comments to the reviews to mark which items you have completed and which ones you cannot implement? This will speed up the review process. It probably will not take long for Fransplaces to render her publication decision once she has gone through the comments and your rebuttals. Many thanks for your patience! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 23:06, 26 March 2025 (UTC) == Mail == {{ygm}} [[User:Serial Number 54129|Serial Number 54129]] ([[User talk:Serial Number 54129|discuss]] • [[Special:Contributions/Serial Number 54129|contribs]]) 12:04, 26 March 2024 (UTC) ==new submissions/need to be imported== Hi, I noticed there are two new submissions (from new editors) at https://en.wikipedia.org/wiki/Wikipedia:WikiJournal_article_nominations, thank you --[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:59, 1 April 2024 (UTC) :I don't have the required permission to import articles from Wikipedia to Wikiversity. I will need the "transwiki importer" permission, presumably to preserve article history and proper copyright attribution. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:40, 15 April 2024 (UTC) ==A message from Guy vandegrift== Hi. I am so-called "founder" of the WikiJournal of Science (although dozens of people contributed much more than I ever did.) I was wondering if the WikiJournal project needs help. If so, let me know.----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 13 April 2024 (UTC) :Yes, I'll email you with the details. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:58, 15 April 2024 (UTC) ::@[[User:Guy vandegrift|Guy vandegrift]] Did you receive the email that I sent last week? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:17, 22 April 2024 (UTC) :::I will look for it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:17, 22 April 2024 (UTC). ::::My guess is that you used the google wikijournal system and it went to a google email I rarely check. I just sent you an email through Wikiversity. Meanwhile I will lookup my google email password and probably find your message.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:35, 22 April 2024 (UTC) == [[WikiJournal_Preprints/Induced_stem_cells]] == Hello, I assume that you are involved in the management of Wikijournals and their preprints. Thank you for your contributions. I'm sending this message to alert you that a preprint is currently subject to copyright-related investigations, this may affect the preprint review procedure and I thought someone who knows more about Wikijournals should be contacted. The background information can be seen at [[Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues]]. In your opinion, what should be done by the custodians for this preprint? I look forward to hearing from you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 6 June 2024 (UTC) :Thanks for bringing this to our attention. What you described is very concerning. We did [[Talk:WikiJournal Preprints/Induced stem cells#Plagiarism check|conduct a plagiarism check]] 3 years ago when the preprint was submitted and it was determined that the similarities were deemed to be common phases in that field. Right now the tool is timing out due to high request volume so I can't do another check now. I'm going to ping @[[User:Evolution and evolvability|Evolution and evolvability]] since he's the handling editor for this submission and he knows more about cells & proteins than me. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:15, 6 June 2024 (UTC) == Question about the WikiJournal license status == Hello. At [[Special:Diff/2639304]], [[User:MGA73]] asked about the Wikijournal license status, so I'm forwarding the question here. Do you know anything about this? Should we contact [[User:Evolution and evolvability]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:02, 30 July 2024 (UTC) == Preprint related to Wikidata == Hello! I have written an article titled "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]". Could you please include this preprint in the list of [[WikiJournal of Science/Potential upcoming articles|Potential upcoming articles]]? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 14:17, 17 February 2025 (UTC) :@[[User:AKA MBG|AKA MBG]] Hello, not sure why I didn't get a notification when you leave this message. I have taken a look at your preprint. Unfortunately I don't think we have the expertise in our editorial board to take on the role for potential publication of your submission. As a general and personal comment, I think you need to tighten up the paper by drawing comparison with existing literature around SPARQL and Wikidata, such as [https://link.springer.com/chapter/10.1007/978-3-319-46547-0_10] and [https://link.springer.com/chapter/10.1007/978-3-031-33455-9_40] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:00, 24 March 2025 (UTC) == Status of WikiJournals == Good morning, I have had an article submitted to WikiJournal PrePrints since October 2024. It seems that the chair of the WikiJournal Usergroup (E&E) is entirely inactive, and I'm not sure what your status is as editor-in-chief of the science journal. If these projects are not currently working, then there should be some kind of alert given so people don't submit articles that will never be reviewed. If they are currently working, please let me know what the next steps in the process are for my submitted article. If there is any way I can help with other articles as well, I am happy to do so. [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 14:00, 6 March 2025 (UTC) :@[[User:Fritzmann2002|Fritzmann2002]] Hello, it has been busy for many of us at the board over the past few months focusing on the grant request and sustainability of the user group, and all of us serving in volunteer capacity with a daytime job. I should have a handling editor for your submission ([[WikiJournal Preprints/Hypericum sechmenii]]) within 2 weeks. Thanks. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:45, 24 March 2025 (UTC) ::@[[User:OhanaUnited|OhanaUnited]], thanks for your response, and apologies for the brusque nature of my original message. I appreciate the work that you do, and want to reiterate my desire to assist in any way that I can! [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 01:45, 25 March 2025 (UTC) == [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]] == Hi OhanaUnited, I'm planning on working on a paper for the WikiJournal of PPB regarding mental health in Sri Lanka (which does not seem to have a corresponding Wikipedia article, so I think this would be a very good start; especially as an aspiring clinical PhD student). I wanted to double check and make sure that this WikiJournal has personnel that can peer-review the article for submission, as there seems to be [[WikiJournal of PPB/Editors|no associate editors]] and the social medias (FB & X accounts) for this specific WikiJournal do not exist [anymore?]. Is this WikiJournal still active and can editors be assigned to my paper once its ready for peer-review? Thank you & thank you to the team for all the work you guys do! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:57, 8 May 2025 (UTC) :Hi, unfortunately I don't have any updates for WikiJournal of PPB on its launch date since the person in charge is on extended absence. I would recommend that you select either WikiJournal of Medicine (since it's mental health) or select another journal with compatible copyright license to publish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:16, 8 May 2025 (UTC) ::I'll work on this paper through the WikiJournal of Medicine then, thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:51, 8 May 2025 (UTC) :::No problem. Thanks for your ongoing support of the journal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:30, 9 May 2025 (UTC) == WikiJournal article nominations == Hi OhanaUnited. More than 5 months ago I have nominated the page [[w:Diffeology|Diffeology]] for submission at the Wikijournal of Science, adding a line at the bottome of the page [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]]. Unfortunately, nobody has created the corresponding preprint at [[WikiJournal Preprints|Wikijournal Preprints]], hence I cannot proceed yet with the formal submission. Since I had already a very positive experience publishing another paper ([[WikiJournal of Science/Poisson manifold|Poisson manifold]]) in the Wikijournal of Science, in the past months I tried, without success, to contact by email the editors who took care of it. I am therefore trying to reach you here. As I wrote also to them, I noticed that at [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]] there are links to several other wikipedia pages which have not been converted to a preprint, despite being many months old. I am therefore wondering if that page is still maintained and with which frequency. This issue was also discussed on [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead]]. I understand that you and the rest of the editorial board has a lot to do and therefore it might be just a matter of waiting. As another user pointed out ([[User talk:OhanaUnited#Status of WikiJournals]]), if there is anything I could do in order to speed up the review process, e.g. creating the preprint page myself, please let me know. In that case (i.e. if the author is allowed to import the page directly from wikipedia), I would suggest to clarify it in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], since these instructions do not specify exactly who is in charge of importing the page. Thanks a lot in advance! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 10:08, 16 September 2025 (UTC) :Hi, an update. @[[User:Marshallsumter|Marshallsumter]] has suggested me in the nomination page to proceed with the import myself. As per our discussion in [[wikipedia:User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints|User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints]], I did attempt to import the page manually at [[WikiJournal Preprints/Diffeology]] and filled in the Authorship declaration form (providing the authors information, suggesting reviewers, etc. and mentioning also that I did the import manually). :One issue is that [[Template:Convert links]] has been deactivated just a few days ago, preventing all the links to other Wikipedia pages to be automatically converted. Since this was the only method written in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], do you know if there are some alternatives, in order to avoid to do it manually? Besides that, I'm also not sure how to make the line "Additional contributors: Wikipedia community" appear under the two names of the authors. :I would appreciate if you or somebody from the editorial board could have a look at these minor issues, so that the review process could start soon. Thanks again! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 22:41, 21 September 2025 (UTC) ::@[[User:Francesco Cattafi|Francesco Cattafi]] Sorry for the late reply. Did the {{tl|Convert links}} end up working again? I see that the links are present. These functions were created long before I joined so I wouldn't be able to troubleshoot them. Sometimes I find that the bugs end up being caused by the most innocent changes in the back end, just like what I encountered [[Wikiversity:Colloquium#Figure numbers are always 1|two weeks ago]]. In the future, if you have some templates or links that aren't working, post a message on [[Wikiversity:Colloquium]] and someone with more knowledge than me may have a solution ready. In related news, there are now two peer review comments which are posted on [[Talk:WikiJournal Preprints/Diffeology]]. I think {{u|Marshallsumter}} is still looking for at least one more peer reviewer. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:01, 9 January 2026 (UTC) :::Hi @[[User:OhanaUnited|OhanaUnited]], thanks for the reply, I didn't know about this Colloquium page. Anyways, the Convert link issue was fixed; I have simply asked the user who deleted that tool to undelete it ([[User_talk:Koavf#Deleting_all_unused_templates]]), so I could use it properly. :::In the coming weeks my coauthor and I will address the two reviewers' comment! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 15:53, 10 January 2026 (UTC) ::::Thanks for your diligence and troubleshoot why it didn't work! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:12, 2 February 2026 (UTC) :::::Hi, just to let you know that we have addressed all three reviewers' comments. Please let us know if any further revisions are needed or if the article will proceed to the next stage of the editorial process. [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 23:09, 3 March 2026 (UTC) ::::::In case you missed it, your article has been published last week and DOI has been issued. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 28 April 2026 (UTC) == Found a potential reviewer == Hello @[[User:OhanaUnited|OhanaUnited]] I hope you are doing well. I write you because some weeks ago, I found a potential reviewer for [[WikiJournal Preprints/Kinematics of the cuboctahedron]] (as we talk about [[Talk:WikiJournal of Science#c-OhanaUnited-20260109204800-Regliste-20260106112200|here]]) and I sent you a mail about it. I'd like to be sure that you indeed received it.<br> On another topic, do you know if there is any progresses on [[WikiJournal Preprints/Pentagram map|my preprint]] ? Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 16:51, 1 February 2026 (UTC) :Thanks for the reminder. I have emailed you about it. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:17, 2 February 2026 (UTC) ::Hello @[[User:OhanaUnited|OhanaUnited]], ::If you have the time, could you answer to my email about the subjects mentioned above, please ? ::Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:21, 3 May 2026 (UTC) :::(Gentle reminder of my previous message.) [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 12:43, 15 May 2026 (UTC) ::::Hello @[[User:OhanaUnited|OhanaUnited]], just to inform you that I replied to the three reviewers comments. I don't know if other reviews are on the way, but in any case I remain available for the continuation of the editorial process. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:46, 16 June 2026 (UTC) ecrbh1d95j4a0z1rma83wiyr139yhlp 2815957 2815956 2026-06-16T14:48:37Z Regliste 3029369 relocation of my last message to a better spot 2815957 wikitext text/x-wiki Thank you for making this happen: [[User:OhanaUnited/Sister Projects Interview]] - I am sure your readers will profit from the better info from all here. Below more info about Wikiversity, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) ==Welcome== '''Hello OhanaUnited, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[Image:Signature_icon.png]] in the edit window makes it simple. To [[Wikiversity:Introduction|get started]], you may <div style="width:50.0%; float:left"> * [[Wikiversity:Guided tour|Take a guided tour]] and learn [[Help:Editing|to edit]]; * Explore our [[Portal:Learning Projects|learning projects]]; * [[Wikiversity:Browse|Browse]] our [[Wikiversity:Portals|portals]], [[Wikiversity:Schools|schools]], and [[Wikiversity:Research|research]] activities; </div> <div style="width:50.0%; float:left"> * Read and help develop our community [[Wikiversity:Policies|policies]]；or * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. </div> <br clear="both"/> And don't forget to [[Wikiversity:Introduction explore|explore]] Wikiversity with the links to your left. [[Wikiversity:Be bold|Be bold]], and see you around Wikiversity! ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) == Environmental experts needed :) == Hi OhanaUnited, There have been a number of environmental projects started here and there... a few I can think of offhand: *[[Project proposal:global warming]] -- I'm not sure where that stands now... it was one of the first proposals back in 2006 I think *[[Bloom Clock]] -- Essentially a phenology project... among other things the data collections will hopefully be handy for later projects tracking changes in bloom time as local and global temperature trends change *[[Radio Discussion/Living on Earth]] -- Something a couple of us were experimenting with this past winter, using a radio show as our "lecture" and collecting materials for further learning. I'm not by any means an expert in environmental science, but as a horticulurist and farmer I'm well-versed in managing my local ecology... let me know if you start something! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 15:06, 28 March 2008 (UTC) :See also [[:Category:Ecology]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 11:18, 29 March 2008 (UTC) == Commons == Is there a page on commons somewhere with the questions? I'm sure I could round up a few interested commonists on IRC if you give me a link :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:15, 30 March 2008 (UTC) == Clarifications == Hi OhanaUnited, I've asked some questions at [[User talk:OhanaUnited/Sister Projects Interview#Voice(s)]] - I'd appreciate if you could clarify before I contribute to your initiative. Thanks, [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:54, 1 April 2008 (UTC) == removing == I removed the signatures after names in order to move forward summarizing the answers... and then I saw that you said to not do that... I reverted... How would be best to summarize the answers? --[[User:Remi|Remi]] 04:05, 21 April 2008 (UTC) :I voiced a related question in the "Voice(s)" section on the talk page.. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:40, 21 April 2008 (UTC) == Publication date == Hi OhanaUnited, would you be able to let us know when [[User:OhanaUnited/Sister_Projects_Interview|your interview]] will be published? Perhaps either on the talk page or on the [[Wikiversity:Colloquium#User:OhanaUnited/Sister Projects Interview - the earliest publication date is April 21|Colloquium]]. Thanks. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:39, 21 April 2008 (UTC) == Font Tag == The font tag is now obsolete. Please adjust your signature to something like: <blockquote> <pre> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] </pre> </blockquote> Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:37, 29 May 2018 (UTC) == Reorganised discussion == This is to let you know that the discussion at [[Talk:WikiJournal User Group#Code of Conduct]] has been reorganised to ease constructive inputs that help in updating the [[WikiJournal User Group/Code of conduct draft|document]]. If you would like to summarily oppose implementation of any Code of Conduct, feel free to place your opposition at [[Talk:WikiJournal User Group#Discussion: Whether any Code of Conduct needs to be defined and implemented]]. For any other constructive inputs please feel free to do so at [[Talk:WikiJournal User Group#Discussion: Salient updates that need to be made to the existing draft]]. Thanks for your cooperation. <span style="font-family:Segoe script">[[w:User:Diptanshu Das|<b style="color:#f00">D</b><b style="color:#f60">ip</b><b style="color:#090">ta</b><b style="color:#00f">ns</b><b style="color:#60c">hu</b>]] [[User talk:Diptanshu Das|&#128172;]]</span> 12:20, 16 December 2018 (UTC) == Maps via Wikidata == I remember you were testing maybe plotting a map of editor locations. I've been testing [https://w.wiki/CGk generating a map in Wikidata]. If we include all journal editors on the WikiJournal's page then it's possible to find the geocoordinates of their employer. Eventually it should be automate-able via [[wikidata:Wikidata:Bot_requests#Automated_addition_of_WikiJournal_metadata_to_Wikidata|this bot request]], but would have to be done manually for now. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:22, 18 November 2019 (UTC) :Note, [https://w.wiki/CWP updated version] with better interface for multiple points. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 23 November 2019 (UTC) == Query at review page == I just noticed there's a query for you at [[Talk:WikiJournal Preprints/Working with Bipolar Disorder During the COVID-19 Pandemic: Both Crisis and Opportunity|this page]] (the editor forgot to ping, or is unaware of the practice). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:43, 17 May 2020 (UTC) == Re: A Phonological Analysis of Selected Nigerian Newscasters Rendition == I appreciate your consideration of my article for publication. However, you have not provided an email address where I could send the word version or preferably, I would like to be guided on how to get the article uploaded on wiki commons. Thank you. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 07:35, 25 August 2022 (UTC) == The Validity of [[WikiJournal Preprints/The Effect of Corticosteroids on the Mortality Rate in COVID-19 Patients, v2]] == Hello Andrew, I'm coming to you to ask whether the mentioned paper's topic/objective is suitable for publication in the WikiJournal of Medicine. I was going to extensively work on it this summer, but I wanted to get written confirmation that this paper would be suited for my time in developing it. I also wanted to see if a Wikijournal of Humanities paper on Meditation would be suitable. I'm not sure if you're familiar with that wikijournal's guidelines, but I figured it was worth asking. Thank you, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:20, 27 August 2022 (UTC) == Request == Please, I do not know whether you could help upload the article if I send its soft copy as MS word document or pdf to you. Thanks. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 03:44, 5 September 2022 (UTC) == Volunteering to help with WikiJournal of Humanities == I kinf of forgot about WikiJournals for a few years, and I am amazed at the progress made. Well, as a real-life professor of sociology, I'd be happy to help with WikiJournal of Humanities which seems to be closed to my field. Do let me know how I can help, assuming of course you need any assistance. (If you reply here, please ping me back, TIA). [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:31, 8 November 2022 (UTC) == In other news == I am a strict believer in learning from the bottoms up (as a teacher who tells students to edit Wikipedia, for example, I never ask them to do things I haven't done myself before). And it so happens, I have a publication that I think is within the scope of WikiJournal Medicine, and now that I know it is indexed in SCOPUS, it meets my university's requirements too. As I am not yet on the board or such, I think I have no COI, so I decided to went ahead and submit my work at [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia]] . Before I finish copyediting it (I think I need to upload images to Wikimedia Commons and reformat references to footnotes) and finish the rest of the submission procedure, can I ask you to confirm that this topic is within the scope of WJMED and our previous conversation does not create any COI for me to submit it (I am fine putting my editorial application fpr WJHUM from yesterday on hold for the duration of the review process, if necessary)? Oh, to confirm, WikiJournals allows and prefers non-anonymous submissions, right? So I don't need to anonymize citations to my own work, etc.? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:08, 10 November 2022 (UTC) :{{re|Piotrus}} Each WikiJournal (Medicine, Science, Humanities) has separate editorial boards, similar to how "Nature Medicine" and "Nature Chemistry" are two different journals, have different editor-in-chief and different ISSN/DOI even though they are both owned and published by Springer Nature. Each WikiJournal operates and makes article decisions independently from each other while sharing same pool of resources (hired contractors, H/R, overhead cost). Therefore, whether or not you are on the Humanities board will not cause a COI when submitting to Medicine. I am the managing editor for Science, so our conversations won't cause any COI. I will defer your question on whether your preprint falls into the scope of Medicine to [[User:Rwatson1955]], who is the managing editor for the Medicine journal. And yes, we [[WikiJournal of Medicine/Publishing#Duties_of_authors|ask that "authors should be given by real names in their articles"]] so there is no need to anonymize. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:58, 10 November 2022 (UTC) ::I submitted [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|my article]] two days ago and filled in a Google Form, which suggested I'd receive confirmation email, but nothing happened and the article still has a notice that it is not submitted for review. Any chance you could check from your end if things are fine or ping someone who can, as maybe I haven't clicked something correctly or such? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:09, 16 November 2022 (UTC) :::{{re|Piotrus}} That's my fault. Been busy with work. I'll process the new submissions today and update the status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:14, 16 November 2022 (UTC) == Concerning an article == Hello, I'm not sure if you are aware that I have written a new article on Wikiversity, entitled: [[WikiJournal Preprints/Orhan Gazi, the first statesman|Orhan Gazi, the first Statesman]], I started it in September 2022 and finished it in March of the same year, and I was hoping that finding some peer reviewers wouldn't take much time. However, the article remained as it was for more than a year, and I had to ask two professors I know personally to check my work, which they did and their notes were sent in pdf format and added [[Talk:WikiJournal Preprints/Orhan Gazi, the first statesman|here]]. Now the article still needs an editor, before it can be finalized and published, and a fellow Wikipedian, [[User:علاء|Alaa]], suggested your name. I hope that perhaps you could check it. Please let me know what you think, best wishes-- [[User:باسم|باسم]] ([[User talk:باسم|discuss]] • [[Special:Contributions/باسم|contribs]]) 20:17, 7 May 2023 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:WikiJournal Bioclogging - ES.pdf]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 15:41, 19 December 2023 (UTC) ==Japanese rendering== Thanks to your help, I could make [[WikiJournal_of_Science/Bioclogging/ja|Japanese translation of bioclogging article]]. I feel that display style of Japanese sentense is wierd, because breakline is restricted to some characters such as "、". Japanese does not break words with spaces, as normal in western languages, and therefore we break lines anywhere. For example, see [[w:ja:バイオクロッギング|Japanese edition of bioclogging article in Wikipedia]]. It can be fixed by using css. For example, in this paragraph バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。 We can set word-break: break-all, and then <span style="word-break: break-all">バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。</span> Setting this to all paragraphs may be a solution. I would like to know if there is a smarter way to do the same thing. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 08:55, 16 February 2024 (UTC) :@[[User:Katsutoshi Seki|Katsutoshi Seki]] Thanks for raising this issue. I can read and write in Chinese (and therefore I can read Japanese Kanji) so I understand what you're describing about the software not finding spaces to break up words to the next line. I have [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBioclogging%2Fja&diff=2606145&oldid=2605982 forced] the software to consider appropriate line break locations. I'm confident with the line breaks in Kanji but less so in Katakana and Hiragana. And I don't know how it may look like under different computer screens (or mobile phone). Please review and see if the line breaks are done accurately. Also, can you please provide a Japanese translation for the phrases "For the English translation, please see this link." and "For the Japanese translation, please see this link."? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:49, 16 February 2024 (UTC) :: Unfortunately, giving <nowiki>{{wbr}}</nowiki> to some places does not help much, because appropriate place for breaking line changes to various width of windows. Therefore, using <nowiki><span style="word-break: break-all"></nowiki> to all paragraphs, as I showed above, is necessary. I would like to know if there is an appropriate way to change the stylesheet in the page at once. For the translation, "For the English translation, please see '''this link'''."　to "英語版は'''このリンク'''参照", and "For the Japanese translation, please see '''this link'''." to "日本語版は'''このリンク'''参照" [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 01:39, 17 February 2024 (UTC) :::Thanks for verifying. I have removed {{tl|wbr}} and added <nowiki><span style="word-break: break-all"></nowiki>. It doesn't seem very effective to bulleted items. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 17 February 2024 (UTC) :::: I also added css to bulleted items. Now it works find. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 04:50, 17 February 2024 (UTC) :::: I created [[Template:BreakAll]] and applied. ChatGPT was helpful for creating the LUA module. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 12:55, 17 February 2024 (UTC) == Article progress == Hi Ohana, it was great to meet you at the conference in November. I finally got around to finishing the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As we discussed, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. Sorry for the very long delay on this article and I apologize if this is not the right forum to report progress. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 03:45, 21 February 2024 (UTC) :Hi @[[User:Buidhe|Buidhe]], our apologies for the very long delay in replying to you. [[User:Fransplace|Fransplace]], the editor-in-chief for WikiJournal of Humanities, will be looking at your submission shortly. Since we already received two reviewers' comments and you have completed your revisions, are you ok with continuing with the submission process? I think we are on the home stretch with very few items remaining. Can you add your comments to the reviews to mark which items you have completed and which ones you cannot implement? This will speed up the review process. It probably will not take long for Fransplaces to render her publication decision once she has gone through the comments and your rebuttals. Many thanks for your patience! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 23:06, 26 March 2025 (UTC) == Mail == {{ygm}} [[User:Serial Number 54129|Serial Number 54129]] ([[User talk:Serial Number 54129|discuss]] • [[Special:Contributions/Serial Number 54129|contribs]]) 12:04, 26 March 2024 (UTC) ==new submissions/need to be imported== Hi, I noticed there are two new submissions (from new editors) at https://en.wikipedia.org/wiki/Wikipedia:WikiJournal_article_nominations, thank you --[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:59, 1 April 2024 (UTC) :I don't have the required permission to import articles from Wikipedia to Wikiversity. I will need the "transwiki importer" permission, presumably to preserve article history and proper copyright attribution. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:40, 15 April 2024 (UTC) ==A message from Guy vandegrift== Hi. I am so-called "founder" of the WikiJournal of Science (although dozens of people contributed much more than I ever did.) I was wondering if the WikiJournal project needs help. If so, let me know.----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 13 April 2024 (UTC) :Yes, I'll email you with the details. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:58, 15 April 2024 (UTC) ::@[[User:Guy vandegrift|Guy vandegrift]] Did you receive the email that I sent last week? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:17, 22 April 2024 (UTC) :::I will look for it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:17, 22 April 2024 (UTC). ::::My guess is that you used the google wikijournal system and it went to a google email I rarely check. I just sent you an email through Wikiversity. Meanwhile I will lookup my google email password and probably find your message.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:35, 22 April 2024 (UTC) == [[WikiJournal_Preprints/Induced_stem_cells]] == Hello, I assume that you are involved in the management of Wikijournals and their preprints. Thank you for your contributions. I'm sending this message to alert you that a preprint is currently subject to copyright-related investigations, this may affect the preprint review procedure and I thought someone who knows more about Wikijournals should be contacted. The background information can be seen at [[Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues]]. In your opinion, what should be done by the custodians for this preprint? I look forward to hearing from you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 6 June 2024 (UTC) :Thanks for bringing this to our attention. What you described is very concerning. We did [[Talk:WikiJournal Preprints/Induced stem cells#Plagiarism check|conduct a plagiarism check]] 3 years ago when the preprint was submitted and it was determined that the similarities were deemed to be common phases in that field. Right now the tool is timing out due to high request volume so I can't do another check now. I'm going to ping @[[User:Evolution and evolvability|Evolution and evolvability]] since he's the handling editor for this submission and he knows more about cells & proteins than me. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:15, 6 June 2024 (UTC) == Question about the WikiJournal license status == Hello. At [[Special:Diff/2639304]], [[User:MGA73]] asked about the Wikijournal license status, so I'm forwarding the question here. Do you know anything about this? Should we contact [[User:Evolution and evolvability]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:02, 30 July 2024 (UTC) == Preprint related to Wikidata == Hello! I have written an article titled "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]". Could you please include this preprint in the list of [[WikiJournal of Science/Potential upcoming articles|Potential upcoming articles]]? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 14:17, 17 February 2025 (UTC) :@[[User:AKA MBG|AKA MBG]] Hello, not sure why I didn't get a notification when you leave this message. I have taken a look at your preprint. Unfortunately I don't think we have the expertise in our editorial board to take on the role for potential publication of your submission. As a general and personal comment, I think you need to tighten up the paper by drawing comparison with existing literature around SPARQL and Wikidata, such as [https://link.springer.com/chapter/10.1007/978-3-319-46547-0_10] and [https://link.springer.com/chapter/10.1007/978-3-031-33455-9_40] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:00, 24 March 2025 (UTC) == Status of WikiJournals == Good morning, I have had an article submitted to WikiJournal PrePrints since October 2024. It seems that the chair of the WikiJournal Usergroup (E&E) is entirely inactive, and I'm not sure what your status is as editor-in-chief of the science journal. If these projects are not currently working, then there should be some kind of alert given so people don't submit articles that will never be reviewed. If they are currently working, please let me know what the next steps in the process are for my submitted article. If there is any way I can help with other articles as well, I am happy to do so. [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 14:00, 6 March 2025 (UTC) :@[[User:Fritzmann2002|Fritzmann2002]] Hello, it has been busy for many of us at the board over the past few months focusing on the grant request and sustainability of the user group, and all of us serving in volunteer capacity with a daytime job. I should have a handling editor for your submission ([[WikiJournal Preprints/Hypericum sechmenii]]) within 2 weeks. Thanks. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:45, 24 March 2025 (UTC) ::@[[User:OhanaUnited|OhanaUnited]], thanks for your response, and apologies for the brusque nature of my original message. I appreciate the work that you do, and want to reiterate my desire to assist in any way that I can! [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 01:45, 25 March 2025 (UTC) == [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]] == Hi OhanaUnited, I'm planning on working on a paper for the WikiJournal of PPB regarding mental health in Sri Lanka (which does not seem to have a corresponding Wikipedia article, so I think this would be a very good start; especially as an aspiring clinical PhD student). I wanted to double check and make sure that this WikiJournal has personnel that can peer-review the article for submission, as there seems to be [[WikiJournal of PPB/Editors|no associate editors]] and the social medias (FB & X accounts) for this specific WikiJournal do not exist [anymore?]. Is this WikiJournal still active and can editors be assigned to my paper once its ready for peer-review? Thank you & thank you to the team for all the work you guys do! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:57, 8 May 2025 (UTC) :Hi, unfortunately I don't have any updates for WikiJournal of PPB on its launch date since the person in charge is on extended absence. I would recommend that you select either WikiJournal of Medicine (since it's mental health) or select another journal with compatible copyright license to publish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:16, 8 May 2025 (UTC) ::I'll work on this paper through the WikiJournal of Medicine then, thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:51, 8 May 2025 (UTC) :::No problem. Thanks for your ongoing support of the journal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:30, 9 May 2025 (UTC) == WikiJournal article nominations == Hi OhanaUnited. More than 5 months ago I have nominated the page [[w:Diffeology|Diffeology]] for submission at the Wikijournal of Science, adding a line at the bottome of the page [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]]. Unfortunately, nobody has created the corresponding preprint at [[WikiJournal Preprints|Wikijournal Preprints]], hence I cannot proceed yet with the formal submission. Since I had already a very positive experience publishing another paper ([[WikiJournal of Science/Poisson manifold|Poisson manifold]]) in the Wikijournal of Science, in the past months I tried, without success, to contact by email the editors who took care of it. I am therefore trying to reach you here. As I wrote also to them, I noticed that at [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]] there are links to several other wikipedia pages which have not been converted to a preprint, despite being many months old. I am therefore wondering if that page is still maintained and with which frequency. This issue was also discussed on [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead]]. I understand that you and the rest of the editorial board has a lot to do and therefore it might be just a matter of waiting. As another user pointed out ([[User talk:OhanaUnited#Status of WikiJournals]]), if there is anything I could do in order to speed up the review process, e.g. creating the preprint page myself, please let me know. In that case (i.e. if the author is allowed to import the page directly from wikipedia), I would suggest to clarify it in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], since these instructions do not specify exactly who is in charge of importing the page. Thanks a lot in advance! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 10:08, 16 September 2025 (UTC) :Hi, an update. @[[User:Marshallsumter|Marshallsumter]] has suggested me in the nomination page to proceed with the import myself. As per our discussion in [[wikipedia:User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints|User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints]], I did attempt to import the page manually at [[WikiJournal Preprints/Diffeology]] and filled in the Authorship declaration form (providing the authors information, suggesting reviewers, etc. and mentioning also that I did the import manually). :One issue is that [[Template:Convert links]] has been deactivated just a few days ago, preventing all the links to other Wikipedia pages to be automatically converted. Since this was the only method written in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], do you know if there are some alternatives, in order to avoid to do it manually? Besides that, I'm also not sure how to make the line "Additional contributors: Wikipedia community" appear under the two names of the authors. :I would appreciate if you or somebody from the editorial board could have a look at these minor issues, so that the review process could start soon. Thanks again! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 22:41, 21 September 2025 (UTC) ::@[[User:Francesco Cattafi|Francesco Cattafi]] Sorry for the late reply. Did the {{tl|Convert links}} end up working again? I see that the links are present. These functions were created long before I joined so I wouldn't be able to troubleshoot them. Sometimes I find that the bugs end up being caused by the most innocent changes in the back end, just like what I encountered [[Wikiversity:Colloquium#Figure numbers are always 1|two weeks ago]]. In the future, if you have some templates or links that aren't working, post a message on [[Wikiversity:Colloquium]] and someone with more knowledge than me may have a solution ready. In related news, there are now two peer review comments which are posted on [[Talk:WikiJournal Preprints/Diffeology]]. I think {{u|Marshallsumter}} is still looking for at least one more peer reviewer. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:01, 9 January 2026 (UTC) :::Hi @[[User:OhanaUnited|OhanaUnited]], thanks for the reply, I didn't know about this Colloquium page. Anyways, the Convert link issue was fixed; I have simply asked the user who deleted that tool to undelete it ([[User_talk:Koavf#Deleting_all_unused_templates]]), so I could use it properly. :::In the coming weeks my coauthor and I will address the two reviewers' comment! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 15:53, 10 January 2026 (UTC) ::::Thanks for your diligence and troubleshoot why it didn't work! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:12, 2 February 2026 (UTC) :::::Hi, just to let you know that we have addressed all three reviewers' comments. Please let us know if any further revisions are needed or if the article will proceed to the next stage of the editorial process. [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 23:09, 3 March 2026 (UTC) ::::::In case you missed it, your article has been published last week and DOI has been issued. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 28 April 2026 (UTC) == Found a potential reviewer == Hello @[[User:OhanaUnited|OhanaUnited]] I hope you are doing well. I write you because some weeks ago, I found a potential reviewer for [[WikiJournal Preprints/Kinematics of the cuboctahedron]] (as we talk about [[Talk:WikiJournal of Science#c-OhanaUnited-20260109204800-Regliste-20260106112200|here]]) and I sent you a mail about it. I'd like to be sure that you indeed received it.<br> On another topic, do you know if there is any progresses on [[WikiJournal Preprints/Pentagram map|my preprint]] ? Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 16:51, 1 February 2026 (UTC) :Thanks for the reminder. I have emailed you about it. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:17, 2 February 2026 (UTC) ::Hello @[[User:OhanaUnited|OhanaUnited]], ::If you have the time, could you answer to my email about the subjects mentioned above, please ? ::Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:21, 3 May 2026 (UTC) :::(Gentle reminder of my previous message.) [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 12:43, 15 May 2026 (UTC) == Answered to reviewers == Hello @[[User:OhanaUnited|OhanaUnited]], just to inform you that I replied to the three reviewers comments. I don't know if other reviews are on the way, but in any case I remain available for the continuation of the editorial process {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:46, 16 June 2026 (UTC) hssdzjl2laeur72g3gxvaax7ind6r23 2815958 2815957 2026-06-16T14:55:06Z OhanaUnited 18921 /* Found a potential reviewer */ Reply 2815958 wikitext text/x-wiki Thank you for making this happen: [[User:OhanaUnited/Sister Projects Interview]] - I am sure your readers will profit from the better info from all here. Below more info about Wikiversity, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) ==Welcome== '''Hello OhanaUnited, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[Image:Signature_icon.png]] in the edit window makes it simple. To [[Wikiversity:Introduction|get started]], you may <div style="width:50.0%; float:left"> * [[Wikiversity:Guided tour|Take a guided tour]] and learn [[Help:Editing|to edit]]; * Explore our [[Portal:Learning Projects|learning projects]]; * [[Wikiversity:Browse|Browse]] our [[Wikiversity:Portals|portals]], [[Wikiversity:Schools|schools]], and [[Wikiversity:Research|research]] activities; </div> <div style="width:50.0%; float:left"> * Read and help develop our community [[Wikiversity:Policies|policies]]；or * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. </div> <br clear="both"/> And don't forget to [[Wikiversity:Introduction explore|explore]] Wikiversity with the links to your left. [[Wikiversity:Be bold|Be bold]], and see you around Wikiversity! ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) == Environmental experts needed :) == Hi OhanaUnited, There have been a number of environmental projects started here and there... a few I can think of offhand: *[[Project proposal:global warming]] -- I'm not sure where that stands now... it was one of the first proposals back in 2006 I think *[[Bloom Clock]] -- Essentially a phenology project... among other things the data collections will hopefully be handy for later projects tracking changes in bloom time as local and global temperature trends change *[[Radio Discussion/Living on Earth]] -- Something a couple of us were experimenting with this past winter, using a radio show as our "lecture" and collecting materials for further learning. I'm not by any means an expert in environmental science, but as a horticulurist and farmer I'm well-versed in managing my local ecology... let me know if you start something! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 15:06, 28 March 2008 (UTC) :See also [[:Category:Ecology]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 11:18, 29 March 2008 (UTC) == Commons == Is there a page on commons somewhere with the questions? I'm sure I could round up a few interested commonists on IRC if you give me a link :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:15, 30 March 2008 (UTC) == Clarifications == Hi OhanaUnited, I've asked some questions at [[User talk:OhanaUnited/Sister Projects Interview#Voice(s)]] - I'd appreciate if you could clarify before I contribute to your initiative. Thanks, [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:54, 1 April 2008 (UTC) == removing == I removed the signatures after names in order to move forward summarizing the answers... and then I saw that you said to not do that... I reverted... How would be best to summarize the answers? --[[User:Remi|Remi]] 04:05, 21 April 2008 (UTC) :I voiced a related question in the "Voice(s)" section on the talk page.. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:40, 21 April 2008 (UTC) == Publication date == Hi OhanaUnited, would you be able to let us know when [[User:OhanaUnited/Sister_Projects_Interview|your interview]] will be published? Perhaps either on the talk page or on the [[Wikiversity:Colloquium#User:OhanaUnited/Sister Projects Interview - the earliest publication date is April 21|Colloquium]]. Thanks. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:39, 21 April 2008 (UTC) == Font Tag == The font tag is now obsolete. Please adjust your signature to something like: <blockquote> <pre> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] </pre> </blockquote> Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:37, 29 May 2018 (UTC) == Reorganised discussion == This is to let you know that the discussion at [[Talk:WikiJournal User Group#Code of Conduct]] has been reorganised to ease constructive inputs that help in updating the [[WikiJournal User Group/Code of conduct draft|document]]. If you would like to summarily oppose implementation of any Code of Conduct, feel free to place your opposition at [[Talk:WikiJournal User Group#Discussion: Whether any Code of Conduct needs to be defined and implemented]]. For any other constructive inputs please feel free to do so at [[Talk:WikiJournal User Group#Discussion: Salient updates that need to be made to the existing draft]]. Thanks for your cooperation. <span style="font-family:Segoe script">[[w:User:Diptanshu Das|<b style="color:#f00">D</b><b style="color:#f60">ip</b><b style="color:#090">ta</b><b style="color:#00f">ns</b><b style="color:#60c">hu</b>]] [[User talk:Diptanshu Das|&#128172;]]</span> 12:20, 16 December 2018 (UTC) == Maps via Wikidata == I remember you were testing maybe plotting a map of editor locations. I've been testing [https://w.wiki/CGk generating a map in Wikidata]. If we include all journal editors on the WikiJournal's page then it's possible to find the geocoordinates of their employer. Eventually it should be automate-able via [[wikidata:Wikidata:Bot_requests#Automated_addition_of_WikiJournal_metadata_to_Wikidata|this bot request]], but would have to be done manually for now. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:22, 18 November 2019 (UTC) :Note, [https://w.wiki/CWP updated version] with better interface for multiple points. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 23 November 2019 (UTC) == Query at review page == I just noticed there's a query for you at [[Talk:WikiJournal Preprints/Working with Bipolar Disorder During the COVID-19 Pandemic: Both Crisis and Opportunity|this page]] (the editor forgot to ping, or is unaware of the practice). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:43, 17 May 2020 (UTC) == Re: A Phonological Analysis of Selected Nigerian Newscasters Rendition == I appreciate your consideration of my article for publication. However, you have not provided an email address where I could send the word version or preferably, I would like to be guided on how to get the article uploaded on wiki commons. Thank you. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 07:35, 25 August 2022 (UTC) == The Validity of [[WikiJournal Preprints/The Effect of Corticosteroids on the Mortality Rate in COVID-19 Patients, v2]] == Hello Andrew, I'm coming to you to ask whether the mentioned paper's topic/objective is suitable for publication in the WikiJournal of Medicine. I was going to extensively work on it this summer, but I wanted to get written confirmation that this paper would be suited for my time in developing it. I also wanted to see if a Wikijournal of Humanities paper on Meditation would be suitable. I'm not sure if you're familiar with that wikijournal's guidelines, but I figured it was worth asking. Thank you, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:20, 27 August 2022 (UTC) == Request == Please, I do not know whether you could help upload the article if I send its soft copy as MS word document or pdf to you. Thanks. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 03:44, 5 September 2022 (UTC) == Volunteering to help with WikiJournal of Humanities == I kinf of forgot about WikiJournals for a few years, and I am amazed at the progress made. Well, as a real-life professor of sociology, I'd be happy to help with WikiJournal of Humanities which seems to be closed to my field. Do let me know how I can help, assuming of course you need any assistance. (If you reply here, please ping me back, TIA). [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:31, 8 November 2022 (UTC) == In other news == I am a strict believer in learning from the bottoms up (as a teacher who tells students to edit Wikipedia, for example, I never ask them to do things I haven't done myself before). And it so happens, I have a publication that I think is within the scope of WikiJournal Medicine, and now that I know it is indexed in SCOPUS, it meets my university's requirements too. As I am not yet on the board or such, I think I have no COI, so I decided to went ahead and submit my work at [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia]] . Before I finish copyediting it (I think I need to upload images to Wikimedia Commons and reformat references to footnotes) and finish the rest of the submission procedure, can I ask you to confirm that this topic is within the scope of WJMED and our previous conversation does not create any COI for me to submit it (I am fine putting my editorial application fpr WJHUM from yesterday on hold for the duration of the review process, if necessary)? Oh, to confirm, WikiJournals allows and prefers non-anonymous submissions, right? So I don't need to anonymize citations to my own work, etc.? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:08, 10 November 2022 (UTC) :{{re|Piotrus}} Each WikiJournal (Medicine, Science, Humanities) has separate editorial boards, similar to how "Nature Medicine" and "Nature Chemistry" are two different journals, have different editor-in-chief and different ISSN/DOI even though they are both owned and published by Springer Nature. Each WikiJournal operates and makes article decisions independently from each other while sharing same pool of resources (hired contractors, H/R, overhead cost). Therefore, whether or not you are on the Humanities board will not cause a COI when submitting to Medicine. I am the managing editor for Science, so our conversations won't cause any COI. I will defer your question on whether your preprint falls into the scope of Medicine to [[User:Rwatson1955]], who is the managing editor for the Medicine journal. And yes, we [[WikiJournal of Medicine/Publishing#Duties_of_authors|ask that "authors should be given by real names in their articles"]] so there is no need to anonymize. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:58, 10 November 2022 (UTC) ::I submitted [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|my article]] two days ago and filled in a Google Form, which suggested I'd receive confirmation email, but nothing happened and the article still has a notice that it is not submitted for review. Any chance you could check from your end if things are fine or ping someone who can, as maybe I haven't clicked something correctly or such? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:09, 16 November 2022 (UTC) :::{{re|Piotrus}} That's my fault. Been busy with work. I'll process the new submissions today and update the status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:14, 16 November 2022 (UTC) == Concerning an article == Hello, I'm not sure if you are aware that I have written a new article on Wikiversity, entitled: [[WikiJournal Preprints/Orhan Gazi, the first statesman|Orhan Gazi, the first Statesman]], I started it in September 2022 and finished it in March of the same year, and I was hoping that finding some peer reviewers wouldn't take much time. However, the article remained as it was for more than a year, and I had to ask two professors I know personally to check my work, which they did and their notes were sent in pdf format and added [[Talk:WikiJournal Preprints/Orhan Gazi, the first statesman|here]]. Now the article still needs an editor, before it can be finalized and published, and a fellow Wikipedian, [[User:علاء|Alaa]], suggested your name. I hope that perhaps you could check it. Please let me know what you think, best wishes-- [[User:باسم|باسم]] ([[User talk:باسم|discuss]] • [[Special:Contributions/باسم|contribs]]) 20:17, 7 May 2023 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:WikiJournal Bioclogging - ES.pdf]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 15:41, 19 December 2023 (UTC) ==Japanese rendering== Thanks to your help, I could make [[WikiJournal_of_Science/Bioclogging/ja|Japanese translation of bioclogging article]]. I feel that display style of Japanese sentense is wierd, because breakline is restricted to some characters such as "、". Japanese does not break words with spaces, as normal in western languages, and therefore we break lines anywhere. For example, see [[w:ja:バイオクロッギング|Japanese edition of bioclogging article in Wikipedia]]. It can be fixed by using css. For example, in this paragraph バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。 We can set word-break: break-all, and then <span style="word-break: break-all">バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。</span> Setting this to all paragraphs may be a solution. I would like to know if there is a smarter way to do the same thing. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 08:55, 16 February 2024 (UTC) :@[[User:Katsutoshi Seki|Katsutoshi Seki]] Thanks for raising this issue. I can read and write in Chinese (and therefore I can read Japanese Kanji) so I understand what you're describing about the software not finding spaces to break up words to the next line. I have [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBioclogging%2Fja&diff=2606145&oldid=2605982 forced] the software to consider appropriate line break locations. I'm confident with the line breaks in Kanji but less so in Katakana and Hiragana. And I don't know how it may look like under different computer screens (or mobile phone). Please review and see if the line breaks are done accurately. Also, can you please provide a Japanese translation for the phrases "For the English translation, please see this link." and "For the Japanese translation, please see this link."? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:49, 16 February 2024 (UTC) :: Unfortunately, giving <nowiki>{{wbr}}</nowiki> to some places does not help much, because appropriate place for breaking line changes to various width of windows. Therefore, using <nowiki><span style="word-break: break-all"></nowiki> to all paragraphs, as I showed above, is necessary. I would like to know if there is an appropriate way to change the stylesheet in the page at once. For the translation, "For the English translation, please see '''this link'''."　to "英語版は'''このリンク'''参照", and "For the Japanese translation, please see '''this link'''." to "日本語版は'''このリンク'''参照" [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 01:39, 17 February 2024 (UTC) :::Thanks for verifying. I have removed {{tl|wbr}} and added <nowiki><span style="word-break: break-all"></nowiki>. It doesn't seem very effective to bulleted items. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 17 February 2024 (UTC) :::: I also added css to bulleted items. Now it works find. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 04:50, 17 February 2024 (UTC) :::: I created [[Template:BreakAll]] and applied. ChatGPT was helpful for creating the LUA module. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 12:55, 17 February 2024 (UTC) == Article progress == Hi Ohana, it was great to meet you at the conference in November. I finally got around to finishing the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As we discussed, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. Sorry for the very long delay on this article and I apologize if this is not the right forum to report progress. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 03:45, 21 February 2024 (UTC) :Hi @[[User:Buidhe|Buidhe]], our apologies for the very long delay in replying to you. [[User:Fransplace|Fransplace]], the editor-in-chief for WikiJournal of Humanities, will be looking at your submission shortly. Since we already received two reviewers' comments and you have completed your revisions, are you ok with continuing with the submission process? I think we are on the home stretch with very few items remaining. Can you add your comments to the reviews to mark which items you have completed and which ones you cannot implement? This will speed up the review process. It probably will not take long for Fransplaces to render her publication decision once she has gone through the comments and your rebuttals. Many thanks for your patience! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 23:06, 26 March 2025 (UTC) == Mail == {{ygm}} [[User:Serial Number 54129|Serial Number 54129]] ([[User talk:Serial Number 54129|discuss]] • [[Special:Contributions/Serial Number 54129|contribs]]) 12:04, 26 March 2024 (UTC) ==new submissions/need to be imported== Hi, I noticed there are two new submissions (from new editors) at https://en.wikipedia.org/wiki/Wikipedia:WikiJournal_article_nominations, thank you --[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:59, 1 April 2024 (UTC) :I don't have the required permission to import articles from Wikipedia to Wikiversity. I will need the "transwiki importer" permission, presumably to preserve article history and proper copyright attribution. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:40, 15 April 2024 (UTC) ==A message from Guy vandegrift== Hi. I am so-called "founder" of the WikiJournal of Science (although dozens of people contributed much more than I ever did.) I was wondering if the WikiJournal project needs help. If so, let me know.----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 13 April 2024 (UTC) :Yes, I'll email you with the details. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:58, 15 April 2024 (UTC) ::@[[User:Guy vandegrift|Guy vandegrift]] Did you receive the email that I sent last week? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:17, 22 April 2024 (UTC) :::I will look for it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:17, 22 April 2024 (UTC). ::::My guess is that you used the google wikijournal system and it went to a google email I rarely check. I just sent you an email through Wikiversity. Meanwhile I will lookup my google email password and probably find your message.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:35, 22 April 2024 (UTC) == [[WikiJournal_Preprints/Induced_stem_cells]] == Hello, I assume that you are involved in the management of Wikijournals and their preprints. Thank you for your contributions. I'm sending this message to alert you that a preprint is currently subject to copyright-related investigations, this may affect the preprint review procedure and I thought someone who knows more about Wikijournals should be contacted. The background information can be seen at [[Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues]]. In your opinion, what should be done by the custodians for this preprint? I look forward to hearing from you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 6 June 2024 (UTC) :Thanks for bringing this to our attention. What you described is very concerning. We did [[Talk:WikiJournal Preprints/Induced stem cells#Plagiarism check|conduct a plagiarism check]] 3 years ago when the preprint was submitted and it was determined that the similarities were deemed to be common phases in that field. Right now the tool is timing out due to high request volume so I can't do another check now. I'm going to ping @[[User:Evolution and evolvability|Evolution and evolvability]] since he's the handling editor for this submission and he knows more about cells & proteins than me. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:15, 6 June 2024 (UTC) == Question about the WikiJournal license status == Hello. At [[Special:Diff/2639304]], [[User:MGA73]] asked about the Wikijournal license status, so I'm forwarding the question here. Do you know anything about this? Should we contact [[User:Evolution and evolvability]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:02, 30 July 2024 (UTC) == Preprint related to Wikidata == Hello! I have written an article titled "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]". Could you please include this preprint in the list of [[WikiJournal of Science/Potential upcoming articles|Potential upcoming articles]]? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 14:17, 17 February 2025 (UTC) :@[[User:AKA MBG|AKA MBG]] Hello, not sure why I didn't get a notification when you leave this message. I have taken a look at your preprint. Unfortunately I don't think we have the expertise in our editorial board to take on the role for potential publication of your submission. As a general and personal comment, I think you need to tighten up the paper by drawing comparison with existing literature around SPARQL and Wikidata, such as [https://link.springer.com/chapter/10.1007/978-3-319-46547-0_10] and [https://link.springer.com/chapter/10.1007/978-3-031-33455-9_40] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:00, 24 March 2025 (UTC) == Status of WikiJournals == Good morning, I have had an article submitted to WikiJournal PrePrints since October 2024. It seems that the chair of the WikiJournal Usergroup (E&E) is entirely inactive, and I'm not sure what your status is as editor-in-chief of the science journal. If these projects are not currently working, then there should be some kind of alert given so people don't submit articles that will never be reviewed. If they are currently working, please let me know what the next steps in the process are for my submitted article. If there is any way I can help with other articles as well, I am happy to do so. [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 14:00, 6 March 2025 (UTC) :@[[User:Fritzmann2002|Fritzmann2002]] Hello, it has been busy for many of us at the board over the past few months focusing on the grant request and sustainability of the user group, and all of us serving in volunteer capacity with a daytime job. I should have a handling editor for your submission ([[WikiJournal Preprints/Hypericum sechmenii]]) within 2 weeks. Thanks. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:45, 24 March 2025 (UTC) ::@[[User:OhanaUnited|OhanaUnited]], thanks for your response, and apologies for the brusque nature of my original message. I appreciate the work that you do, and want to reiterate my desire to assist in any way that I can! [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 01:45, 25 March 2025 (UTC) == [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]] == Hi OhanaUnited, I'm planning on working on a paper for the WikiJournal of PPB regarding mental health in Sri Lanka (which does not seem to have a corresponding Wikipedia article, so I think this would be a very good start; especially as an aspiring clinical PhD student). I wanted to double check and make sure that this WikiJournal has personnel that can peer-review the article for submission, as there seems to be [[WikiJournal of PPB/Editors|no associate editors]] and the social medias (FB & X accounts) for this specific WikiJournal do not exist [anymore?]. Is this WikiJournal still active and can editors be assigned to my paper once its ready for peer-review? Thank you & thank you to the team for all the work you guys do! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:57, 8 May 2025 (UTC) :Hi, unfortunately I don't have any updates for WikiJournal of PPB on its launch date since the person in charge is on extended absence. I would recommend that you select either WikiJournal of Medicine (since it's mental health) or select another journal with compatible copyright license to publish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:16, 8 May 2025 (UTC) ::I'll work on this paper through the WikiJournal of Medicine then, thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:51, 8 May 2025 (UTC) :::No problem. Thanks for your ongoing support of the journal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:30, 9 May 2025 (UTC) == WikiJournal article nominations == Hi OhanaUnited. More than 5 months ago I have nominated the page [[w:Diffeology|Diffeology]] for submission at the Wikijournal of Science, adding a line at the bottome of the page [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]]. Unfortunately, nobody has created the corresponding preprint at [[WikiJournal Preprints|Wikijournal Preprints]], hence I cannot proceed yet with the formal submission. Since I had already a very positive experience publishing another paper ([[WikiJournal of Science/Poisson manifold|Poisson manifold]]) in the Wikijournal of Science, in the past months I tried, without success, to contact by email the editors who took care of it. I am therefore trying to reach you here. As I wrote also to them, I noticed that at [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]] there are links to several other wikipedia pages which have not been converted to a preprint, despite being many months old. I am therefore wondering if that page is still maintained and with which frequency. This issue was also discussed on [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead]]. I understand that you and the rest of the editorial board has a lot to do and therefore it might be just a matter of waiting. As another user pointed out ([[User talk:OhanaUnited#Status of WikiJournals]]), if there is anything I could do in order to speed up the review process, e.g. creating the preprint page myself, please let me know. In that case (i.e. if the author is allowed to import the page directly from wikipedia), I would suggest to clarify it in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], since these instructions do not specify exactly who is in charge of importing the page. Thanks a lot in advance! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 10:08, 16 September 2025 (UTC) :Hi, an update. @[[User:Marshallsumter|Marshallsumter]] has suggested me in the nomination page to proceed with the import myself. As per our discussion in [[wikipedia:User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints|User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints]], I did attempt to import the page manually at [[WikiJournal Preprints/Diffeology]] and filled in the Authorship declaration form (providing the authors information, suggesting reviewers, etc. and mentioning also that I did the import manually). :One issue is that [[Template:Convert links]] has been deactivated just a few days ago, preventing all the links to other Wikipedia pages to be automatically converted. Since this was the only method written in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], do you know if there are some alternatives, in order to avoid to do it manually? Besides that, I'm also not sure how to make the line "Additional contributors: Wikipedia community" appear under the two names of the authors. :I would appreciate if you or somebody from the editorial board could have a look at these minor issues, so that the review process could start soon. Thanks again! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 22:41, 21 September 2025 (UTC) ::@[[User:Francesco Cattafi|Francesco Cattafi]] Sorry for the late reply. Did the {{tl|Convert links}} end up working again? I see that the links are present. These functions were created long before I joined so I wouldn't be able to troubleshoot them. Sometimes I find that the bugs end up being caused by the most innocent changes in the back end, just like what I encountered [[Wikiversity:Colloquium#Figure numbers are always 1|two weeks ago]]. In the future, if you have some templates or links that aren't working, post a message on [[Wikiversity:Colloquium]] and someone with more knowledge than me may have a solution ready. In related news, there are now two peer review comments which are posted on [[Talk:WikiJournal Preprints/Diffeology]]. I think {{u|Marshallsumter}} is still looking for at least one more peer reviewer. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:01, 9 January 2026 (UTC) :::Hi @[[User:OhanaUnited|OhanaUnited]], thanks for the reply, I didn't know about this Colloquium page. Anyways, the Convert link issue was fixed; I have simply asked the user who deleted that tool to undelete it ([[User_talk:Koavf#Deleting_all_unused_templates]]), so I could use it properly. :::In the coming weeks my coauthor and I will address the two reviewers' comment! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 15:53, 10 January 2026 (UTC) ::::Thanks for your diligence and troubleshoot why it didn't work! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:12, 2 February 2026 (UTC) :::::Hi, just to let you know that we have addressed all three reviewers' comments. Please let us know if any further revisions are needed or if the article will proceed to the next stage of the editorial process. [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 23:09, 3 March 2026 (UTC) ::::::In case you missed it, your article has been published last week and DOI has been issued. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 28 April 2026 (UTC) == Found a potential reviewer == Hello @[[User:OhanaUnited|OhanaUnited]] I hope you are doing well. I write you because some weeks ago, I found a potential reviewer for [[WikiJournal Preprints/Kinematics of the cuboctahedron]] (as we talk about [[Talk:WikiJournal of Science#c-OhanaUnited-20260109204800-Regliste-20260106112200|here]]) and I sent you a mail about it. I'd like to be sure that you indeed received it.<br> On another topic, do you know if there is any progresses on [[WikiJournal Preprints/Pentagram map|my preprint]] ? Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 16:51, 1 February 2026 (UTC) :Thanks for the reminder. I have emailed you about it. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:17, 2 February 2026 (UTC) ::Hello @[[User:OhanaUnited|OhanaUnited]], ::If you have the time, could you answer to my email about the subjects mentioned above, please ? ::Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:21, 3 May 2026 (UTC) :::(Gentle reminder of my previous message.) [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 12:43, 15 May 2026 (UTC) ::::@[[User:Regliste|Regliste]] Only one reviewer accepted my invite to peer review Kinematics of the cuboctahedron. I will send another batch of review invitations later this week. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:55, 16 June 2026 (UTC) == Answered to reviewers == Hello @[[User:OhanaUnited|OhanaUnited]], just to inform you that I replied to the three reviewers comments. I don't know if other reviews are on the way, but in any case I remain available for the continuation of the editorial process {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:46, 16 June 2026 (UTC) 97likphdjkc1qcl7acobfesqnsnv6t0 2815959 2815958 2026-06-16T14:56:56Z OhanaUnited 18921 /* Answered to reviewers */ Reply 2815959 wikitext text/x-wiki Thank you for making this happen: [[User:OhanaUnited/Sister Projects Interview]] - I am sure your readers will profit from the better info from all here. Below more info about Wikiversity, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) ==Welcome== '''Hello OhanaUnited, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[Image:Signature_icon.png]] in the edit window makes it simple. To [[Wikiversity:Introduction|get started]], you may <div style="width:50.0%; float:left"> * [[Wikiversity:Guided tour|Take a guided tour]] and learn [[Help:Editing|to edit]]; * Explore our [[Portal:Learning Projects|learning projects]]; * [[Wikiversity:Browse|Browse]] our [[Wikiversity:Portals|portals]], [[Wikiversity:Schools|schools]], and [[Wikiversity:Research|research]] activities; </div> <div style="width:50.0%; float:left"> * Read and help develop our community [[Wikiversity:Policies|policies]]；or * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. </div> <br clear="both"/> And don't forget to [[Wikiversity:Introduction explore|explore]] Wikiversity with the links to your left. [[Wikiversity:Be bold|Be bold]], and see you around Wikiversity! ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) == Environmental experts needed :) == Hi OhanaUnited, There have been a number of environmental projects started here and there... a few I can think of offhand: *[[Project proposal:global warming]] -- I'm not sure where that stands now... it was one of the first proposals back in 2006 I think *[[Bloom Clock]] -- Essentially a phenology project... among other things the data collections will hopefully be handy for later projects tracking changes in bloom time as local and global temperature trends change *[[Radio Discussion/Living on Earth]] -- Something a couple of us were experimenting with this past winter, using a radio show as our "lecture" and collecting materials for further learning. I'm not by any means an expert in environmental science, but as a horticulurist and farmer I'm well-versed in managing my local ecology... let me know if you start something! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 15:06, 28 March 2008 (UTC) :See also [[:Category:Ecology]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 11:18, 29 March 2008 (UTC) == Commons == Is there a page on commons somewhere with the questions? I'm sure I could round up a few interested commonists on IRC if you give me a link :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:15, 30 March 2008 (UTC) == Clarifications == Hi OhanaUnited, I've asked some questions at [[User talk:OhanaUnited/Sister Projects Interview#Voice(s)]] - I'd appreciate if you could clarify before I contribute to your initiative. Thanks, [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:54, 1 April 2008 (UTC) == removing == I removed the signatures after names in order to move forward summarizing the answers... and then I saw that you said to not do that... I reverted... How would be best to summarize the answers? --[[User:Remi|Remi]] 04:05, 21 April 2008 (UTC) :I voiced a related question in the "Voice(s)" section on the talk page.. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:40, 21 April 2008 (UTC) == Publication date == Hi OhanaUnited, would you be able to let us know when [[User:OhanaUnited/Sister_Projects_Interview|your interview]] will be published? Perhaps either on the talk page or on the [[Wikiversity:Colloquium#User:OhanaUnited/Sister Projects Interview - the earliest publication date is April 21|Colloquium]]. Thanks. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:39, 21 April 2008 (UTC) == Font Tag == The font tag is now obsolete. Please adjust your signature to something like: <blockquote> <pre> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] </pre> </blockquote> Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:37, 29 May 2018 (UTC) == Reorganised discussion == This is to let you know that the discussion at [[Talk:WikiJournal User Group#Code of Conduct]] has been reorganised to ease constructive inputs that help in updating the [[WikiJournal User Group/Code of conduct draft|document]]. If you would like to summarily oppose implementation of any Code of Conduct, feel free to place your opposition at [[Talk:WikiJournal User Group#Discussion: Whether any Code of Conduct needs to be defined and implemented]]. For any other constructive inputs please feel free to do so at [[Talk:WikiJournal User Group#Discussion: Salient updates that need to be made to the existing draft]]. Thanks for your cooperation. <span style="font-family:Segoe script">[[w:User:Diptanshu Das|<b style="color:#f00">D</b><b style="color:#f60">ip</b><b style="color:#090">ta</b><b style="color:#00f">ns</b><b style="color:#60c">hu</b>]] [[User talk:Diptanshu Das|&#128172;]]</span> 12:20, 16 December 2018 (UTC) == Maps via Wikidata == I remember you were testing maybe plotting a map of editor locations. I've been testing [https://w.wiki/CGk generating a map in Wikidata]. If we include all journal editors on the WikiJournal's page then it's possible to find the geocoordinates of their employer. Eventually it should be automate-able via [[wikidata:Wikidata:Bot_requests#Automated_addition_of_WikiJournal_metadata_to_Wikidata|this bot request]], but would have to be done manually for now. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:22, 18 November 2019 (UTC) :Note, [https://w.wiki/CWP updated version] with better interface for multiple points. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 23 November 2019 (UTC) == Query at review page == I just noticed there's a query for you at [[Talk:WikiJournal Preprints/Working with Bipolar Disorder During the COVID-19 Pandemic: Both Crisis and Opportunity|this page]] (the editor forgot to ping, or is unaware of the practice). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:43, 17 May 2020 (UTC) == Re: A Phonological Analysis of Selected Nigerian Newscasters Rendition == I appreciate your consideration of my article for publication. However, you have not provided an email address where I could send the word version or preferably, I would like to be guided on how to get the article uploaded on wiki commons. Thank you. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 07:35, 25 August 2022 (UTC) == The Validity of [[WikiJournal Preprints/The Effect of Corticosteroids on the Mortality Rate in COVID-19 Patients, v2]] == Hello Andrew, I'm coming to you to ask whether the mentioned paper's topic/objective is suitable for publication in the WikiJournal of Medicine. I was going to extensively work on it this summer, but I wanted to get written confirmation that this paper would be suited for my time in developing it. I also wanted to see if a Wikijournal of Humanities paper on Meditation would be suitable. I'm not sure if you're familiar with that wikijournal's guidelines, but I figured it was worth asking. Thank you, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:20, 27 August 2022 (UTC) == Request == Please, I do not know whether you could help upload the article if I send its soft copy as MS word document or pdf to you. Thanks. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 03:44, 5 September 2022 (UTC) == Volunteering to help with WikiJournal of Humanities == I kinf of forgot about WikiJournals for a few years, and I am amazed at the progress made. Well, as a real-life professor of sociology, I'd be happy to help with WikiJournal of Humanities which seems to be closed to my field. Do let me know how I can help, assuming of course you need any assistance. (If you reply here, please ping me back, TIA). [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:31, 8 November 2022 (UTC) == In other news == I am a strict believer in learning from the bottoms up (as a teacher who tells students to edit Wikipedia, for example, I never ask them to do things I haven't done myself before). And it so happens, I have a publication that I think is within the scope of WikiJournal Medicine, and now that I know it is indexed in SCOPUS, it meets my university's requirements too. As I am not yet on the board or such, I think I have no COI, so I decided to went ahead and submit my work at [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia]] . Before I finish copyediting it (I think I need to upload images to Wikimedia Commons and reformat references to footnotes) and finish the rest of the submission procedure, can I ask you to confirm that this topic is within the scope of WJMED and our previous conversation does not create any COI for me to submit it (I am fine putting my editorial application fpr WJHUM from yesterday on hold for the duration of the review process, if necessary)? Oh, to confirm, WikiJournals allows and prefers non-anonymous submissions, right? So I don't need to anonymize citations to my own work, etc.? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:08, 10 November 2022 (UTC) :{{re|Piotrus}} Each WikiJournal (Medicine, Science, Humanities) has separate editorial boards, similar to how "Nature Medicine" and "Nature Chemistry" are two different journals, have different editor-in-chief and different ISSN/DOI even though they are both owned and published by Springer Nature. Each WikiJournal operates and makes article decisions independently from each other while sharing same pool of resources (hired contractors, H/R, overhead cost). Therefore, whether or not you are on the Humanities board will not cause a COI when submitting to Medicine. I am the managing editor for Science, so our conversations won't cause any COI. I will defer your question on whether your preprint falls into the scope of Medicine to [[User:Rwatson1955]], who is the managing editor for the Medicine journal. And yes, we [[WikiJournal of Medicine/Publishing#Duties_of_authors|ask that "authors should be given by real names in their articles"]] so there is no need to anonymize. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:58, 10 November 2022 (UTC) ::I submitted [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|my article]] two days ago and filled in a Google Form, which suggested I'd receive confirmation email, but nothing happened and the article still has a notice that it is not submitted for review. Any chance you could check from your end if things are fine or ping someone who can, as maybe I haven't clicked something correctly or such? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:09, 16 November 2022 (UTC) :::{{re|Piotrus}} That's my fault. Been busy with work. I'll process the new submissions today and update the status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:14, 16 November 2022 (UTC) == Concerning an article == Hello, I'm not sure if you are aware that I have written a new article on Wikiversity, entitled: [[WikiJournal Preprints/Orhan Gazi, the first statesman|Orhan Gazi, the first Statesman]], I started it in September 2022 and finished it in March of the same year, and I was hoping that finding some peer reviewers wouldn't take much time. However, the article remained as it was for more than a year, and I had to ask two professors I know personally to check my work, which they did and their notes were sent in pdf format and added [[Talk:WikiJournal Preprints/Orhan Gazi, the first statesman|here]]. Now the article still needs an editor, before it can be finalized and published, and a fellow Wikipedian, [[User:علاء|Alaa]], suggested your name. I hope that perhaps you could check it. Please let me know what you think, best wishes-- [[User:باسم|باسم]] ([[User talk:باسم|discuss]] • [[Special:Contributions/باسم|contribs]]) 20:17, 7 May 2023 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:WikiJournal Bioclogging - ES.pdf]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 15:41, 19 December 2023 (UTC) ==Japanese rendering== Thanks to your help, I could make [[WikiJournal_of_Science/Bioclogging/ja|Japanese translation of bioclogging article]]. I feel that display style of Japanese sentense is wierd, because breakline is restricted to some characters such as "、". Japanese does not break words with spaces, as normal in western languages, and therefore we break lines anywhere. For example, see [[w:ja:バイオクロッギング|Japanese edition of bioclogging article in Wikipedia]]. It can be fixed by using css. For example, in this paragraph バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。 We can set word-break: break-all, and then <span style="word-break: break-all">バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。</span> Setting this to all paragraphs may be a solution. I would like to know if there is a smarter way to do the same thing. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 08:55, 16 February 2024 (UTC) :@[[User:Katsutoshi Seki|Katsutoshi Seki]] Thanks for raising this issue. I can read and write in Chinese (and therefore I can read Japanese Kanji) so I understand what you're describing about the software not finding spaces to break up words to the next line. I have [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBioclogging%2Fja&diff=2606145&oldid=2605982 forced] the software to consider appropriate line break locations. I'm confident with the line breaks in Kanji but less so in Katakana and Hiragana. And I don't know how it may look like under different computer screens (or mobile phone). Please review and see if the line breaks are done accurately. Also, can you please provide a Japanese translation for the phrases "For the English translation, please see this link." and "For the Japanese translation, please see this link."? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:49, 16 February 2024 (UTC) :: Unfortunately, giving <nowiki>{{wbr}}</nowiki> to some places does not help much, because appropriate place for breaking line changes to various width of windows. Therefore, using <nowiki><span style="word-break: break-all"></nowiki> to all paragraphs, as I showed above, is necessary. I would like to know if there is an appropriate way to change the stylesheet in the page at once. For the translation, "For the English translation, please see '''this link'''."　to "英語版は'''このリンク'''参照", and "For the Japanese translation, please see '''this link'''." to "日本語版は'''このリンク'''参照" [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 01:39, 17 February 2024 (UTC) :::Thanks for verifying. I have removed {{tl|wbr}} and added <nowiki><span style="word-break: break-all"></nowiki>. It doesn't seem very effective to bulleted items. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 17 February 2024 (UTC) :::: I also added css to bulleted items. Now it works find. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 04:50, 17 February 2024 (UTC) :::: I created [[Template:BreakAll]] and applied. ChatGPT was helpful for creating the LUA module. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 12:55, 17 February 2024 (UTC) == Article progress == Hi Ohana, it was great to meet you at the conference in November. I finally got around to finishing the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As we discussed, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. Sorry for the very long delay on this article and I apologize if this is not the right forum to report progress. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 03:45, 21 February 2024 (UTC) :Hi @[[User:Buidhe|Buidhe]], our apologies for the very long delay in replying to you. [[User:Fransplace|Fransplace]], the editor-in-chief for WikiJournal of Humanities, will be looking at your submission shortly. Since we already received two reviewers' comments and you have completed your revisions, are you ok with continuing with the submission process? I think we are on the home stretch with very few items remaining. Can you add your comments to the reviews to mark which items you have completed and which ones you cannot implement? This will speed up the review process. It probably will not take long for Fransplaces to render her publication decision once she has gone through the comments and your rebuttals. Many thanks for your patience! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 23:06, 26 March 2025 (UTC) == Mail == {{ygm}} [[User:Serial Number 54129|Serial Number 54129]] ([[User talk:Serial Number 54129|discuss]] • [[Special:Contributions/Serial Number 54129|contribs]]) 12:04, 26 March 2024 (UTC) ==new submissions/need to be imported== Hi, I noticed there are two new submissions (from new editors) at https://en.wikipedia.org/wiki/Wikipedia:WikiJournal_article_nominations, thank you --[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:59, 1 April 2024 (UTC) :I don't have the required permission to import articles from Wikipedia to Wikiversity. I will need the "transwiki importer" permission, presumably to preserve article history and proper copyright attribution. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:40, 15 April 2024 (UTC) ==A message from Guy vandegrift== Hi. I am so-called "founder" of the WikiJournal of Science (although dozens of people contributed much more than I ever did.) I was wondering if the WikiJournal project needs help. If so, let me know.----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 13 April 2024 (UTC) :Yes, I'll email you with the details. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:58, 15 April 2024 (UTC) ::@[[User:Guy vandegrift|Guy vandegrift]] Did you receive the email that I sent last week? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:17, 22 April 2024 (UTC) :::I will look for it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:17, 22 April 2024 (UTC). ::::My guess is that you used the google wikijournal system and it went to a google email I rarely check. I just sent you an email through Wikiversity. Meanwhile I will lookup my google email password and probably find your message.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:35, 22 April 2024 (UTC) == [[WikiJournal_Preprints/Induced_stem_cells]] == Hello, I assume that you are involved in the management of Wikijournals and their preprints. Thank you for your contributions. I'm sending this message to alert you that a preprint is currently subject to copyright-related investigations, this may affect the preprint review procedure and I thought someone who knows more about Wikijournals should be contacted. The background information can be seen at [[Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues]]. In your opinion, what should be done by the custodians for this preprint? I look forward to hearing from you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 6 June 2024 (UTC) :Thanks for bringing this to our attention. What you described is very concerning. We did [[Talk:WikiJournal Preprints/Induced stem cells#Plagiarism check|conduct a plagiarism check]] 3 years ago when the preprint was submitted and it was determined that the similarities were deemed to be common phases in that field. Right now the tool is timing out due to high request volume so I can't do another check now. I'm going to ping @[[User:Evolution and evolvability|Evolution and evolvability]] since he's the handling editor for this submission and he knows more about cells & proteins than me. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:15, 6 June 2024 (UTC) == Question about the WikiJournal license status == Hello. At [[Special:Diff/2639304]], [[User:MGA73]] asked about the Wikijournal license status, so I'm forwarding the question here. Do you know anything about this? Should we contact [[User:Evolution and evolvability]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:02, 30 July 2024 (UTC) == Preprint related to Wikidata == Hello! I have written an article titled "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]". Could you please include this preprint in the list of [[WikiJournal of Science/Potential upcoming articles|Potential upcoming articles]]? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 14:17, 17 February 2025 (UTC) :@[[User:AKA MBG|AKA MBG]] Hello, not sure why I didn't get a notification when you leave this message. I have taken a look at your preprint. Unfortunately I don't think we have the expertise in our editorial board to take on the role for potential publication of your submission. As a general and personal comment, I think you need to tighten up the paper by drawing comparison with existing literature around SPARQL and Wikidata, such as [https://link.springer.com/chapter/10.1007/978-3-319-46547-0_10] and [https://link.springer.com/chapter/10.1007/978-3-031-33455-9_40] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:00, 24 March 2025 (UTC) == Status of WikiJournals == Good morning, I have had an article submitted to WikiJournal PrePrints since October 2024. It seems that the chair of the WikiJournal Usergroup (E&E) is entirely inactive, and I'm not sure what your status is as editor-in-chief of the science journal. If these projects are not currently working, then there should be some kind of alert given so people don't submit articles that will never be reviewed. If they are currently working, please let me know what the next steps in the process are for my submitted article. If there is any way I can help with other articles as well, I am happy to do so. [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 14:00, 6 March 2025 (UTC) :@[[User:Fritzmann2002|Fritzmann2002]] Hello, it has been busy for many of us at the board over the past few months focusing on the grant request and sustainability of the user group, and all of us serving in volunteer capacity with a daytime job. I should have a handling editor for your submission ([[WikiJournal Preprints/Hypericum sechmenii]]) within 2 weeks. Thanks. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:45, 24 March 2025 (UTC) ::@[[User:OhanaUnited|OhanaUnited]], thanks for your response, and apologies for the brusque nature of my original message. I appreciate the work that you do, and want to reiterate my desire to assist in any way that I can! [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 01:45, 25 March 2025 (UTC) == [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]] == Hi OhanaUnited, I'm planning on working on a paper for the WikiJournal of PPB regarding mental health in Sri Lanka (which does not seem to have a corresponding Wikipedia article, so I think this would be a very good start; especially as an aspiring clinical PhD student). I wanted to double check and make sure that this WikiJournal has personnel that can peer-review the article for submission, as there seems to be [[WikiJournal of PPB/Editors|no associate editors]] and the social medias (FB & X accounts) for this specific WikiJournal do not exist [anymore?]. Is this WikiJournal still active and can editors be assigned to my paper once its ready for peer-review? Thank you & thank you to the team for all the work you guys do! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:57, 8 May 2025 (UTC) :Hi, unfortunately I don't have any updates for WikiJournal of PPB on its launch date since the person in charge is on extended absence. I would recommend that you select either WikiJournal of Medicine (since it's mental health) or select another journal with compatible copyright license to publish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:16, 8 May 2025 (UTC) ::I'll work on this paper through the WikiJournal of Medicine then, thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:51, 8 May 2025 (UTC) :::No problem. Thanks for your ongoing support of the journal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:30, 9 May 2025 (UTC) == WikiJournal article nominations == Hi OhanaUnited. More than 5 months ago I have nominated the page [[w:Diffeology|Diffeology]] for submission at the Wikijournal of Science, adding a line at the bottome of the page [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]]. Unfortunately, nobody has created the corresponding preprint at [[WikiJournal Preprints|Wikijournal Preprints]], hence I cannot proceed yet with the formal submission. Since I had already a very positive experience publishing another paper ([[WikiJournal of Science/Poisson manifold|Poisson manifold]]) in the Wikijournal of Science, in the past months I tried, without success, to contact by email the editors who took care of it. I am therefore trying to reach you here. As I wrote also to them, I noticed that at [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]] there are links to several other wikipedia pages which have not been converted to a preprint, despite being many months old. I am therefore wondering if that page is still maintained and with which frequency. This issue was also discussed on [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead]]. I understand that you and the rest of the editorial board has a lot to do and therefore it might be just a matter of waiting. As another user pointed out ([[User talk:OhanaUnited#Status of WikiJournals]]), if there is anything I could do in order to speed up the review process, e.g. creating the preprint page myself, please let me know. In that case (i.e. if the author is allowed to import the page directly from wikipedia), I would suggest to clarify it in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], since these instructions do not specify exactly who is in charge of importing the page. Thanks a lot in advance! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 10:08, 16 September 2025 (UTC) :Hi, an update. @[[User:Marshallsumter|Marshallsumter]] has suggested me in the nomination page to proceed with the import myself. As per our discussion in [[wikipedia:User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints|User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints]], I did attempt to import the page manually at [[WikiJournal Preprints/Diffeology]] and filled in the Authorship declaration form (providing the authors information, suggesting reviewers, etc. and mentioning also that I did the import manually). :One issue is that [[Template:Convert links]] has been deactivated just a few days ago, preventing all the links to other Wikipedia pages to be automatically converted. Since this was the only method written in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], do you know if there are some alternatives, in order to avoid to do it manually? Besides that, I'm also not sure how to make the line "Additional contributors: Wikipedia community" appear under the two names of the authors. :I would appreciate if you or somebody from the editorial board could have a look at these minor issues, so that the review process could start soon. Thanks again! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 22:41, 21 September 2025 (UTC) ::@[[User:Francesco Cattafi|Francesco Cattafi]] Sorry for the late reply. Did the {{tl|Convert links}} end up working again? I see that the links are present. These functions were created long before I joined so I wouldn't be able to troubleshoot them. Sometimes I find that the bugs end up being caused by the most innocent changes in the back end, just like what I encountered [[Wikiversity:Colloquium#Figure numbers are always 1|two weeks ago]]. In the future, if you have some templates or links that aren't working, post a message on [[Wikiversity:Colloquium]] and someone with more knowledge than me may have a solution ready. In related news, there are now two peer review comments which are posted on [[Talk:WikiJournal Preprints/Diffeology]]. I think {{u|Marshallsumter}} is still looking for at least one more peer reviewer. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:01, 9 January 2026 (UTC) :::Hi @[[User:OhanaUnited|OhanaUnited]], thanks for the reply, I didn't know about this Colloquium page. Anyways, the Convert link issue was fixed; I have simply asked the user who deleted that tool to undelete it ([[User_talk:Koavf#Deleting_all_unused_templates]]), so I could use it properly. :::In the coming weeks my coauthor and I will address the two reviewers' comment! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 15:53, 10 January 2026 (UTC) ::::Thanks for your diligence and troubleshoot why it didn't work! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:12, 2 February 2026 (UTC) :::::Hi, just to let you know that we have addressed all three reviewers' comments. Please let us know if any further revisions are needed or if the article will proceed to the next stage of the editorial process. [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 23:09, 3 March 2026 (UTC) ::::::In case you missed it, your article has been published last week and DOI has been issued. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 28 April 2026 (UTC) == Found a potential reviewer == Hello @[[User:OhanaUnited|OhanaUnited]] I hope you are doing well. I write you because some weeks ago, I found a potential reviewer for [[WikiJournal Preprints/Kinematics of the cuboctahedron]] (as we talk about [[Talk:WikiJournal of Science#c-OhanaUnited-20260109204800-Regliste-20260106112200|here]]) and I sent you a mail about it. I'd like to be sure that you indeed received it.<br> On another topic, do you know if there is any progresses on [[WikiJournal Preprints/Pentagram map|my preprint]] ? Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 16:51, 1 February 2026 (UTC) :Thanks for the reminder. I have emailed you about it. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:17, 2 February 2026 (UTC) ::Hello @[[User:OhanaUnited|OhanaUnited]], ::If you have the time, could you answer to my email about the subjects mentioned above, please ? ::Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:21, 3 May 2026 (UTC) :::(Gentle reminder of my previous message.) [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 12:43, 15 May 2026 (UTC) ::::@[[User:Regliste|Regliste]] Only one reviewer accepted my invite to peer review Kinematics of the cuboctahedron. I will send another batch of review invitations later this week. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:55, 16 June 2026 (UTC) == Answered to reviewers == Hello @[[User:OhanaUnited|OhanaUnited]], just to inform you that I replied to the three reviewers comments. I don't know if other reviews are on the way, but in any case I remain available for the continuation of the editorial process {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:46, 16 June 2026 (UTC) :I have one more review for Pentagram map that I'm expecting, but the review is not due for another 6 days. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:56, 16 June 2026 (UTC) h7juz64lg2m0zt7kco2px0icoun3cno 2815960 2815959 2026-06-16T14:58:43Z Regliste 3029369 /* Answered to reviewers */ Reply 2815960 wikitext text/x-wiki Thank you for making this happen: [[User:OhanaUnited/Sister Projects Interview]] - I am sure your readers will profit from the better info from all here. Below more info about Wikiversity, ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) ==Welcome== '''Hello OhanaUnited, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity:What is Wikiversity?|Wikiversity]]!''' If you need [[Help:Contents|help]], feel free to visit my talk page, or [[Wikiversity:Contact|contact us]] and [[Wikiversity:Questions|ask questions]]. After you leave a comment on a [[Wikiversity:Talk page|talk page]], remember to [[Wikiversity:Signature|sign and date]]; it helps everyone follow the threads of the discussion. The signature icon [[Image:Signature_icon.png]] in the edit window makes it simple. To [[Wikiversity:Introduction|get started]], you may <div style="width:50.0%; float:left"> * [[Wikiversity:Guided tour|Take a guided tour]] and learn [[Help:Editing|to edit]]; * Explore our [[Portal:Learning Projects|learning projects]]; * [[Wikiversity:Browse|Browse]] our [[Wikiversity:Portals|portals]], [[Wikiversity:Schools|schools]], and [[Wikiversity:Research|research]] activities; </div> <div style="width:50.0%; float:left"> * Read and help develop our community [[Wikiversity:Policies|policies]]；or * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [irc://irc.freenode.net/wikiversity-en <kbd>#wikiversity-en</kbd>]. </div> <br clear="both"/> And don't forget to [[Wikiversity:Introduction explore|explore]] Wikiversity with the links to your left. [[Wikiversity:Be bold|Be bold]], and see you around Wikiversity! ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 18:51, 27 March 2008 (UTC) == Environmental experts needed :) == Hi OhanaUnited, There have been a number of environmental projects started here and there... a few I can think of offhand: *[[Project proposal:global warming]] -- I'm not sure where that stands now... it was one of the first proposals back in 2006 I think *[[Bloom Clock]] -- Essentially a phenology project... among other things the data collections will hopefully be handy for later projects tracking changes in bloom time as local and global temperature trends change *[[Radio Discussion/Living on Earth]] -- Something a couple of us were experimenting with this past winter, using a radio show as our "lecture" and collecting materials for further learning. I'm not by any means an expert in environmental science, but as a horticulurist and farmer I'm well-versed in managing my local ecology... let me know if you start something! --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 15:06, 28 March 2008 (UTC) :See also [[:Category:Ecology]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 11:18, 29 March 2008 (UTC) == Commons == Is there a page on commons somewhere with the questions? I'm sure I could round up a few interested commonists on IRC if you give me a link :). --[[User:SB_Johnny|{{font|color=green|'''SB_Johnny'''}}]] | ^{[[User_talk:SB_Johnny|{{font|color=green|talk}}]]} 14:15, 30 March 2008 (UTC) == Clarifications == Hi OhanaUnited, I've asked some questions at [[User talk:OhanaUnited/Sister Projects Interview#Voice(s)]] - I'd appreciate if you could clarify before I contribute to your initiative. Thanks, [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:54, 1 April 2008 (UTC) == removing == I removed the signatures after names in order to move forward summarizing the answers... and then I saw that you said to not do that... I reverted... How would be best to summarize the answers? --[[User:Remi|Remi]] 04:05, 21 April 2008 (UTC) :I voiced a related question in the "Voice(s)" section on the talk page.. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:40, 21 April 2008 (UTC) == Publication date == Hi OhanaUnited, would you be able to let us know when [[User:OhanaUnited/Sister_Projects_Interview|your interview]] will be published? Perhaps either on the talk page or on the [[Wikiversity:Colloquium#User:OhanaUnited/Sister Projects Interview - the earliest publication date is April 21|Colloquium]]. Thanks. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 12:39, 21 April 2008 (UTC) == Font Tag == The font tag is now obsolete. Please adjust your signature to something like: <blockquote> <pre> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] </pre> </blockquote> Let me know if you have any questions. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 17:37, 29 May 2018 (UTC) == Reorganised discussion == This is to let you know that the discussion at [[Talk:WikiJournal User Group#Code of Conduct]] has been reorganised to ease constructive inputs that help in updating the [[WikiJournal User Group/Code of conduct draft|document]]. If you would like to summarily oppose implementation of any Code of Conduct, feel free to place your opposition at [[Talk:WikiJournal User Group#Discussion: Whether any Code of Conduct needs to be defined and implemented]]. For any other constructive inputs please feel free to do so at [[Talk:WikiJournal User Group#Discussion: Salient updates that need to be made to the existing draft]]. Thanks for your cooperation. <span style="font-family:Segoe script">[[w:User:Diptanshu Das|<b style="color:#f00">D</b><b style="color:#f60">ip</b><b style="color:#090">ta</b><b style="color:#00f">ns</b><b style="color:#60c">hu</b>]] [[User talk:Diptanshu Das|&#128172;]]</span> 12:20, 16 December 2018 (UTC) == Maps via Wikidata == I remember you were testing maybe plotting a map of editor locations. I've been testing [https://w.wiki/CGk generating a map in Wikidata]. If we include all journal editors on the WikiJournal's page then it's possible to find the geocoordinates of their employer. Eventually it should be automate-able via [[wikidata:Wikidata:Bot_requests#Automated_addition_of_WikiJournal_metadata_to_Wikidata|this bot request]], but would have to be done manually for now. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:22, 18 November 2019 (UTC) :Note, [https://w.wiki/CWP updated version] with better interface for multiple points. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:48, 23 November 2019 (UTC) == Query at review page == I just noticed there's a query for you at [[Talk:WikiJournal Preprints/Working with Bipolar Disorder During the COVID-19 Pandemic: Both Crisis and Opportunity|this page]] (the editor forgot to ping, or is unaware of the practice). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:43, 17 May 2020 (UTC) == Re: A Phonological Analysis of Selected Nigerian Newscasters Rendition == I appreciate your consideration of my article for publication. However, you have not provided an email address where I could send the word version or preferably, I would like to be guided on how to get the article uploaded on wiki commons. Thank you. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 07:35, 25 August 2022 (UTC) == The Validity of [[WikiJournal Preprints/The Effect of Corticosteroids on the Mortality Rate in COVID-19 Patients, v2]] == Hello Andrew, I'm coming to you to ask whether the mentioned paper's topic/objective is suitable for publication in the WikiJournal of Medicine. I was going to extensively work on it this summer, but I wanted to get written confirmation that this paper would be suited for my time in developing it. I also wanted to see if a Wikijournal of Humanities paper on Meditation would be suitable. I'm not sure if you're familiar with that wikijournal's guidelines, but I figured it was worth asking. Thank you, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:20, 27 August 2022 (UTC) == Request == Please, I do not know whether you could help upload the article if I send its soft copy as MS word document or pdf to you. Thanks. [[User:Margob28|Margob28]] ([[User talk:Margob28|discuss]] • [[Special:Contributions/Margob28|contribs]]) 03:44, 5 September 2022 (UTC) == Volunteering to help with WikiJournal of Humanities == I kinf of forgot about WikiJournals for a few years, and I am amazed at the progress made. Well, as a real-life professor of sociology, I'd be happy to help with WikiJournal of Humanities which seems to be closed to my field. Do let me know how I can help, assuming of course you need any assistance. (If you reply here, please ping me back, TIA). [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:31, 8 November 2022 (UTC) == In other news == I am a strict believer in learning from the bottoms up (as a teacher who tells students to edit Wikipedia, for example, I never ask them to do things I haven't done myself before). And it so happens, I have a publication that I think is within the scope of WikiJournal Medicine, and now that I know it is indexed in SCOPUS, it meets my university's requirements too. As I am not yet on the board or such, I think I have no COI, so I decided to went ahead and submit my work at [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia]] . Before I finish copyediting it (I think I need to upload images to Wikimedia Commons and reformat references to footnotes) and finish the rest of the submission procedure, can I ask you to confirm that this topic is within the scope of WJMED and our previous conversation does not create any COI for me to submit it (I am fine putting my editorial application fpr WJHUM from yesterday on hold for the duration of the review process, if necessary)? Oh, to confirm, WikiJournals allows and prefers non-anonymous submissions, right? So I don't need to anonymize citations to my own work, etc.? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 05:08, 10 November 2022 (UTC) :{{re|Piotrus}} Each WikiJournal (Medicine, Science, Humanities) has separate editorial boards, similar to how "Nature Medicine" and "Nature Chemistry" are two different journals, have different editor-in-chief and different ISSN/DOI even though they are both owned and published by Springer Nature. Each WikiJournal operates and makes article decisions independently from each other while sharing same pool of resources (hired contractors, H/R, overhead cost). Therefore, whether or not you are on the Humanities board will not cause a COI when submitting to Medicine. I am the managing editor for Science, so our conversations won't cause any COI. I will defer your question on whether your preprint falls into the scope of Medicine to [[User:Rwatson1955]], who is the managing editor for the Medicine journal. And yes, we [[WikiJournal of Medicine/Publishing#Duties_of_authors|ask that "authors should be given by real names in their articles"]] so there is no need to anonymize. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:58, 10 November 2022 (UTC) ::I submitted [[WikiJournal Preprints/Where experts and amateurs meet: the ideological hobby of medical volunteering on Wikipedia|my article]] two days ago and filled in a Google Form, which suggested I'd receive confirmation email, but nothing happened and the article still has a notice that it is not submitted for review. Any chance you could check from your end if things are fine or ping someone who can, as maybe I haven't clicked something correctly or such? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 14:09, 16 November 2022 (UTC) :::{{re|Piotrus}} That's my fault. Been busy with work. I'll process the new submissions today and update the status. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:14, 16 November 2022 (UTC) == Concerning an article == Hello, I'm not sure if you are aware that I have written a new article on Wikiversity, entitled: [[WikiJournal Preprints/Orhan Gazi, the first statesman|Orhan Gazi, the first Statesman]], I started it in September 2022 and finished it in March of the same year, and I was hoping that finding some peer reviewers wouldn't take much time. However, the article remained as it was for more than a year, and I had to ask two professors I know personally to check my work, which they did and their notes were sent in pdf format and added [[Talk:WikiJournal Preprints/Orhan Gazi, the first statesman|here]]. Now the article still needs an editor, before it can be finalized and published, and a fellow Wikipedian, [[User:علاء|Alaa]], suggested your name. I hope that perhaps you could check it. Please let me know what you think, best wishes-- [[User:باسم|باسم]] ([[User talk:باسم|discuss]] • [[Special:Contributions/باسم|contribs]]) 20:17, 7 May 2023 (UTC) == Files Missing Information == Thanks for uploading files to Wikiversity. All files must have source and license information to stay at Wikiversity. The following files are missing {{tlx|Information}} and/or [[Wikiversity:License tags]], and will be deleted if the missing information is not added. See [[Wikiversity:Uploading files]] for more information. {{colbegin|3}} * [[:File:WikiJournal Bioclogging - ES.pdf]] {{colend}} [[User:MaintenanceBot|MaintenanceBot]] ([[User talk:MaintenanceBot|discuss]] • [[Special:Contributions/MaintenanceBot|contribs]]) 15:41, 19 December 2023 (UTC) ==Japanese rendering== Thanks to your help, I could make [[WikiJournal_of_Science/Bioclogging/ja|Japanese translation of bioclogging article]]. I feel that display style of Japanese sentense is wierd, because breakline is restricted to some characters such as "、". Japanese does not break words with spaces, as normal in western languages, and therefore we break lines anywhere. For example, see [[w:ja:バイオクロッギング|Japanese edition of bioclogging article in Wikipedia]]. It can be fixed by using css. For example, in this paragraph バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。 We can set word-break: break-all, and then <span style="word-break: break-all">バイオクロッギングは、水が浸透する様々な現場で観察される。たとえば、[[w:ja:ため池|ため池]]、浸透トレンチ、[[w:ja:灌漑|灌漑]]水路、[[w:ja:下水処理場|下水処理場]]、人工湿地、廃棄物処分場における遮水ライナー、川床や土壌のような自然環境などである。また、透過反応壁 ([[:w:Permeable reactive barrier|PRB]]) や微生物利用石油増進回収法 ([[:w:Microbial enhanced oil recovery|MEOR]]) などにおいて、[[w:ja:帯水層|帯水層]]における[[w:ja:地下水|地下水]]の流れにも影響を及ぼす。適度な水の浸透速度を保つことが必要とされるような現場では、バイオクロッギングが問題となり、定期的に水を抜くなどの対策が取られることがある。一方で、たとえば、難透水層を作って浸透速度を低下させたり、地盤工学的性質を改善させたりするなど、バイオクロッギングが有効に活用されることもある。</span> Setting this to all paragraphs may be a solution. I would like to know if there is a smarter way to do the same thing. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 08:55, 16 February 2024 (UTC) :@[[User:Katsutoshi Seki|Katsutoshi Seki]] Thanks for raising this issue. I can read and write in Chinese (and therefore I can read Japanese Kanji) so I understand what you're describing about the software not finding spaces to break up words to the next line. I have [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBioclogging%2Fja&diff=2606145&oldid=2605982 forced] the software to consider appropriate line break locations. I'm confident with the line breaks in Kanji but less so in Katakana and Hiragana. And I don't know how it may look like under different computer screens (or mobile phone). Please review and see if the line breaks are done accurately. Also, can you please provide a Japanese translation for the phrases "For the English translation, please see this link." and "For the Japanese translation, please see this link."? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:49, 16 February 2024 (UTC) :: Unfortunately, giving <nowiki>{{wbr}}</nowiki> to some places does not help much, because appropriate place for breaking line changes to various width of windows. Therefore, using <nowiki><span style="word-break: break-all"></nowiki> to all paragraphs, as I showed above, is necessary. I would like to know if there is an appropriate way to change the stylesheet in the page at once. For the translation, "For the English translation, please see '''this link'''."　to "英語版は'''このリンク'''参照", and "For the Japanese translation, please see '''this link'''." to "日本語版は'''このリンク'''参照" [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 01:39, 17 February 2024 (UTC) :::Thanks for verifying. I have removed {{tl|wbr}} and added <nowiki><span style="word-break: break-all"></nowiki>. It doesn't seem very effective to bulleted items. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 17 February 2024 (UTC) :::: I also added css to bulleted items. Now it works find. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 04:50, 17 February 2024 (UTC) :::: I created [[Template:BreakAll]] and applied. ChatGPT was helpful for creating the LUA module. [[User:Katsutoshi Seki|Katsutoshi Seki]] ([[User talk:Katsutoshi Seki|discuss]] • [[Special:Contributions/Katsutoshi Seki|contribs]]) 12:55, 17 February 2024 (UTC) == Article progress == Hi Ohana, it was great to meet you at the conference in November. I finally got around to finishing the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As we discussed, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. Sorry for the very long delay on this article and I apologize if this is not the right forum to report progress. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 03:45, 21 February 2024 (UTC) :Hi @[[User:Buidhe|Buidhe]], our apologies for the very long delay in replying to you. [[User:Fransplace|Fransplace]], the editor-in-chief for WikiJournal of Humanities, will be looking at your submission shortly. Since we already received two reviewers' comments and you have completed your revisions, are you ok with continuing with the submission process? I think we are on the home stretch with very few items remaining. Can you add your comments to the reviews to mark which items you have completed and which ones you cannot implement? This will speed up the review process. It probably will not take long for Fransplaces to render her publication decision once she has gone through the comments and your rebuttals. Many thanks for your patience! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 23:06, 26 March 2025 (UTC) == Mail == {{ygm}} [[User:Serial Number 54129|Serial Number 54129]] ([[User talk:Serial Number 54129|discuss]] • [[Special:Contributions/Serial Number 54129|contribs]]) 12:04, 26 March 2024 (UTC) ==new submissions/need to be imported== Hi, I noticed there are two new submissions (from new editors) at https://en.wikipedia.org/wiki/Wikipedia:WikiJournal_article_nominations, thank you --[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:59, 1 April 2024 (UTC) :I don't have the required permission to import articles from Wikipedia to Wikiversity. I will need the "transwiki importer" permission, presumably to preserve article history and proper copyright attribution. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:40, 15 April 2024 (UTC) ==A message from Guy vandegrift== Hi. I am so-called "founder" of the WikiJournal of Science (although dozens of people contributed much more than I ever did.) I was wondering if the WikiJournal project needs help. If so, let me know.----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:25, 13 April 2024 (UTC) :Yes, I'll email you with the details. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:58, 15 April 2024 (UTC) ::@[[User:Guy vandegrift|Guy vandegrift]] Did you receive the email that I sent last week? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 18:17, 22 April 2024 (UTC) :::I will look for it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:17, 22 April 2024 (UTC). ::::My guess is that you used the google wikijournal system and it went to a google email I rarely check. I just sent you an email through Wikiversity. Meanwhile I will lookup my google email password and probably find your message.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:35, 22 April 2024 (UTC) == [[WikiJournal_Preprints/Induced_stem_cells]] == Hello, I assume that you are involved in the management of Wikijournals and their preprints. Thank you for your contributions. I'm sending this message to alert you that a preprint is currently subject to copyright-related investigations, this may affect the preprint review procedure and I thought someone who knows more about Wikijournals should be contacted. The background information can be seen at [[Wikiversity:Request_custodian_action#Induced_stem_cells_copyright_issues]]. In your opinion, what should be done by the custodians for this preprint? I look forward to hearing from you. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:02, 6 June 2024 (UTC) :Thanks for bringing this to our attention. What you described is very concerning. We did [[Talk:WikiJournal Preprints/Induced stem cells#Plagiarism check|conduct a plagiarism check]] 3 years ago when the preprint was submitted and it was determined that the similarities were deemed to be common phases in that field. Right now the tool is timing out due to high request volume so I can't do another check now. I'm going to ping @[[User:Evolution and evolvability|Evolution and evolvability]] since he's the handling editor for this submission and he knows more about cells & proteins than me. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:15, 6 June 2024 (UTC) == Question about the WikiJournal license status == Hello. At [[Special:Diff/2639304]], [[User:MGA73]] asked about the Wikijournal license status, so I'm forwarding the question here. Do you know anything about this? Should we contact [[User:Evolution and evolvability]]? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:02, 30 July 2024 (UTC) == Preprint related to Wikidata == Hello! I have written an article titled "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]". Could you please include this preprint in the list of [[WikiJournal of Science/Potential upcoming articles|Potential upcoming articles]]? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 14:17, 17 February 2025 (UTC) :@[[User:AKA MBG|AKA MBG]] Hello, not sure why I didn't get a notification when you leave this message. I have taken a look at your preprint. Unfortunately I don't think we have the expertise in our editorial board to take on the role for potential publication of your submission. As a general and personal comment, I think you need to tighten up the paper by drawing comparison with existing literature around SPARQL and Wikidata, such as [https://link.springer.com/chapter/10.1007/978-3-319-46547-0_10] and [https://link.springer.com/chapter/10.1007/978-3-031-33455-9_40] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:00, 24 March 2025 (UTC) == Status of WikiJournals == Good morning, I have had an article submitted to WikiJournal PrePrints since October 2024. It seems that the chair of the WikiJournal Usergroup (E&E) is entirely inactive, and I'm not sure what your status is as editor-in-chief of the science journal. If these projects are not currently working, then there should be some kind of alert given so people don't submit articles that will never be reviewed. If they are currently working, please let me know what the next steps in the process are for my submitted article. If there is any way I can help with other articles as well, I am happy to do so. [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 14:00, 6 March 2025 (UTC) :@[[User:Fritzmann2002|Fritzmann2002]] Hello, it has been busy for many of us at the board over the past few months focusing on the grant request and sustainability of the user group, and all of us serving in volunteer capacity with a daytime job. I should have a handling editor for your submission ([[WikiJournal Preprints/Hypericum sechmenii]]) within 2 weeks. Thanks. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:45, 24 March 2025 (UTC) ::@[[User:OhanaUnited|OhanaUnited]], thanks for your response, and apologies for the brusque nature of my original message. I appreciate the work that you do, and want to reiterate my desire to assist in any way that I can! [[User:Fritzmann2002|Fritzmann2002]] ([[User talk:Fritzmann2002|discuss]] • [[Special:Contributions/Fritzmann2002|contribs]]) 01:45, 25 March 2025 (UTC) == [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]] == Hi OhanaUnited, I'm planning on working on a paper for the WikiJournal of PPB regarding mental health in Sri Lanka (which does not seem to have a corresponding Wikipedia article, so I think this would be a very good start; especially as an aspiring clinical PhD student). I wanted to double check and make sure that this WikiJournal has personnel that can peer-review the article for submission, as there seems to be [[WikiJournal of PPB/Editors|no associate editors]] and the social medias (FB & X accounts) for this specific WikiJournal do not exist [anymore?]. Is this WikiJournal still active and can editors be assigned to my paper once its ready for peer-review? Thank you & thank you to the team for all the work you guys do! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:57, 8 May 2025 (UTC) :Hi, unfortunately I don't have any updates for WikiJournal of PPB on its launch date since the person in charge is on extended absence. I would recommend that you select either WikiJournal of Medicine (since it's mental health) or select another journal with compatible copyright license to publish. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:16, 8 May 2025 (UTC) ::I'll work on this paper through the WikiJournal of Medicine then, thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:51, 8 May 2025 (UTC) :::No problem. Thanks for your ongoing support of the journal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:30, 9 May 2025 (UTC) == WikiJournal article nominations == Hi OhanaUnited. More than 5 months ago I have nominated the page [[w:Diffeology|Diffeology]] for submission at the Wikijournal of Science, adding a line at the bottome of the page [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]]. Unfortunately, nobody has created the corresponding preprint at [[WikiJournal Preprints|Wikijournal Preprints]], hence I cannot proceed yet with the formal submission. Since I had already a very positive experience publishing another paper ([[WikiJournal of Science/Poisson manifold|Poisson manifold]]) in the Wikijournal of Science, in the past months I tried, without success, to contact by email the editors who took care of it. I am therefore trying to reach you here. As I wrote also to them, I noticed that at [[w:Wikipedia:WikiJournal article nominations|Wikipedia:WikiJournal article nominations]] there are links to several other wikipedia pages which have not been converted to a preprint, despite being many months old. I am therefore wondering if that page is still maintained and with which frequency. This issue was also discussed on [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead]]. I understand that you and the rest of the editorial board has a lot to do and therefore it might be just a matter of waiting. As another user pointed out ([[User talk:OhanaUnited#Status of WikiJournals]]), if there is anything I could do in order to speed up the review process, e.g. creating the preprint page myself, please let me know. In that case (i.e. if the author is allowed to import the page directly from wikipedia), I would suggest to clarify it in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], since these instructions do not specify exactly who is in charge of importing the page. Thanks a lot in advance! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 10:08, 16 September 2025 (UTC) :Hi, an update. @[[User:Marshallsumter|Marshallsumter]] has suggested me in the nomination page to proceed with the import myself. As per our discussion in [[wikipedia:User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints|User_talk:Marshallsumter#Importing_Wikipedia_articles_to_Wikipreprints]], I did attempt to import the page manually at [[WikiJournal Preprints/Diffeology]] and filled in the Authorship declaration form (providing the authors information, suggesting reviewers, etc. and mentioning also that I did the import manually). :One issue is that [[Template:Convert links]] has been deactivated just a few days ago, preventing all the links to other Wikipedia pages to be automatically converted. Since this was the only method written in [[WikiJournal User Group/Editorial guidelines#Importing from Wikipedia]], do you know if there are some alternatives, in order to avoid to do it manually? Besides that, I'm also not sure how to make the line "Additional contributors: Wikipedia community" appear under the two names of the authors. :I would appreciate if you or somebody from the editorial board could have a look at these minor issues, so that the review process could start soon. Thanks again! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 22:41, 21 September 2025 (UTC) ::@[[User:Francesco Cattafi|Francesco Cattafi]] Sorry for the late reply. Did the {{tl|Convert links}} end up working again? I see that the links are present. These functions were created long before I joined so I wouldn't be able to troubleshoot them. Sometimes I find that the bugs end up being caused by the most innocent changes in the back end, just like what I encountered [[Wikiversity:Colloquium#Figure numbers are always 1|two weeks ago]]. In the future, if you have some templates or links that aren't working, post a message on [[Wikiversity:Colloquium]] and someone with more knowledge than me may have a solution ready. In related news, there are now two peer review comments which are posted on [[Talk:WikiJournal Preprints/Diffeology]]. I think {{u|Marshallsumter}} is still looking for at least one more peer reviewer. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:01, 9 January 2026 (UTC) :::Hi @[[User:OhanaUnited|OhanaUnited]], thanks for the reply, I didn't know about this Colloquium page. Anyways, the Convert link issue was fixed; I have simply asked the user who deleted that tool to undelete it ([[User_talk:Koavf#Deleting_all_unused_templates]]), so I could use it properly. :::In the coming weeks my coauthor and I will address the two reviewers' comment! [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 15:53, 10 January 2026 (UTC) ::::Thanks for your diligence and troubleshoot why it didn't work! [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:12, 2 February 2026 (UTC) :::::Hi, just to let you know that we have addressed all three reviewers' comments. Please let us know if any further revisions are needed or if the article will proceed to the next stage of the editorial process. [[User:Francesco Cattafi|Francesco Cattafi]] ([[User talk:Francesco Cattafi|discuss]] • [[Special:Contributions/Francesco Cattafi|contribs]]) 23:09, 3 March 2026 (UTC) ::::::In case you missed it, your article has been published last week and DOI has been issued. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:47, 28 April 2026 (UTC) == Found a potential reviewer == Hello @[[User:OhanaUnited|OhanaUnited]] I hope you are doing well. I write you because some weeks ago, I found a potential reviewer for [[WikiJournal Preprints/Kinematics of the cuboctahedron]] (as we talk about [[Talk:WikiJournal of Science#c-OhanaUnited-20260109204800-Regliste-20260106112200|here]]) and I sent you a mail about it. I'd like to be sure that you indeed received it.<br> On another topic, do you know if there is any progresses on [[WikiJournal Preprints/Pentagram map|my preprint]] ? Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 16:51, 1 February 2026 (UTC) :Thanks for the reminder. I have emailed you about it. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 21:17, 2 February 2026 (UTC) ::Hello @[[User:OhanaUnited|OhanaUnited]], ::If you have the time, could you answer to my email about the subjects mentioned above, please ? ::Best regards, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:21, 3 May 2026 (UTC) :::(Gentle reminder of my previous message.) [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 12:43, 15 May 2026 (UTC) ::::@[[User:Regliste|Regliste]] Only one reviewer accepted my invite to peer review Kinematics of the cuboctahedron. I will send another batch of review invitations later this week. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:55, 16 June 2026 (UTC) == Answered to reviewers == Hello @[[User:OhanaUnited|OhanaUnited]], just to inform you that I replied to the three reviewers comments. I don't know if other reviews are on the way, but in any case I remain available for the continuation of the editorial process {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:46, 16 June 2026 (UTC) :I have one more review for Pentagram map that I'm expecting, but the review is not due for another 6 days. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 14:56, 16 June 2026 (UTC) ::Alright, there is no hurry at all. Thank you for the information. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:58, 16 June 2026 (UTC) 5ys14awd91kzxy3t4kdhwy3u3vp395r 0 55819 2816002 1818129 2026-06-16T19:07:53Z Taylor 49 2929747 PNG->SVG fixing wrong uppercase "N" 2816002 wikitext text/x-wiki {{Educational Media Awareness Campaign/POTD|Nuclear Fission|Kernspaltung -- induced nuclear fission.svg|Diagram showing the nuclear fission of Uranium, producing Rubidium and Cesium, free neutrons and energy. The exact products of nuclear fusion are variable; common fission products include zirconium, xenon and palladium.| [[:commons:Category:Nuclear Power|Nuclear Power images]]- [[:commons:Category:Energy|Energy images]]<br>[[:commons:Category:Mechanics|Mechanics images]] - [[:commons:Category:Electromagnetism|Electromagnetism images]] - [[:commons:Category:Optics|Optics images]] - [[:commons:Category:Atomic physics|Atomic physics images]] <br>[[:commons:Category:Physics|Images relating to physics in general]]}} qp1fhgm5goc23eztihrl4lywngifo3j 2816003 2816002 2026-06-16T19:08:46Z Taylor 49 2929747 qq 2816003 wikitext text/x-wiki {{Educational Media Awareness Campaign/POTD|Nuclear Fission|Kernspaltung -- induced nuclear fission.svg|Diagram showing the nuclear fission of Uranium, producing Barium and Krypton, free neutrons and energy. The exact products of nuclear fusion are variable; common fission products include zirconium, xenon and palladium.| [[:commons:Category:Nuclear Power|Nuclear Power images]]- [[:commons:Category:Energy|Energy images]]<br>[[:commons:Category:Mechanics|Mechanics images]] - [[:commons:Category:Electromagnetism|Electromagnetism images]] - [[:commons:Category:Optics|Optics images]] - [[:commons:Category:Atomic physics|Atomic physics images]] <br>[[:commons:Category:Physics|Images relating to physics in general]]}} pdy3kxzq9gy40mdrstiuvnrzjyb2uk2 4 56709 2816052 2324576 2026-06-17T00:35:11Z Jtneill 10242 /* The edit toolbar */ Update image 2816052 wikitext text/x-wiki <includeonly>__NOEDITSECTION__</includeonly> ==The edit toolbar== [[File:Modern Editing Toolbar for WikiPedia.png|thumb|center|800px|Wikiversity edit toolbar.]] * When you '''[[Wikiversity:Introduction|edit pages]]''' there is an edit toolbar at the top of the edit window. The toolbar provides access to common editing tasks such as creating '''bold text'''. * If you are new to [[wiki]], just type and add plain text. Your knowledge and ideas are the most important contribution to Wikiversity. Other Wikiversity participants will help to format your text. In fact, since this ''is'' a wiki, you should expect all of your contributions to be altered and improved by others! * For more advanced information about editing wiki pages, such as creating [[Help:Link|hypertext links]], see [[Introduction to Wikiversity]] (for new wiki participants) and [[Help:Editing|Editing help]] (for more advanced participants). ==Start a new Wikiversity webpage== [[Image:EditURL.png|thumb|right|400px|One way to create a new page is by typing its name into a web browser.]] * An easy way to create a new Wikiversity page is to think of a descriptive name for a topic and then type it into the window of your web browser that shows page URLs. Just add the new page name after,<BR><code><nowiki>http://en.wikiversity.org/wiki/</nowiki></code> and hit your enter or return key. * Additional help is available about [[Help:Starting a new page|creating new pages]]. [[Image:Edit this page.png|thumb|right|322px|If the page name doesn't exist, you will be prompted to edit the page and create it!]] ==If your "new" page already exists== * If the page name you select is already in use you will be shown the existing page. * It is good practice to use '''"search"''' before creating a new page. You may find that there is already an existing page about the topic you are interested in. ==What to say on your new page== * Wikiversity accepts '''''educational content'''''. Just start typing! * For additional information see [[Wikiversity:Adding content|Adding content]]. * Experiment with editing in the [[Wikiversity:Sandbox|Sandbox]] __NOTOC__ lp7b6utrefl9pd2hj814fe41cls5t0m 0 60970 2816078 2071805 2026-06-17T05:55:57Z Jtneill 10242 Adjust categories 2816078 wikitext text/x-wiki ==All psychology [[journals]]== * [http://lamp.infosys.deakin.edu.au/era/?page=jfordet15&selfor=17 ERA FOR Code 17: Psychology] * [[w:List of psychology journals|List of psychology journals]] (Wikipedia) ==[[Open access]] psychology [[journals]]== This section lists open psychology journals and their rankings. * [https://www.scimagojr.com/journalrank.php?category=3201&openaccess=true scimagojr] * [https://www.doaj.org/search?source=%7B%22query%22%3A%7B%22filtered%22%3A%7B%22filter%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22term%22%3A%7B%22index.classification.exact%22%3A%22Psychology%22%7D%7D%2C%7B%22term%22%3A%7B%22_type%22%3A%22journal%22%7D%7D%5D%7D%7D%2C%22query%22%3A%7B%22match_all%22%3A%7B%7D%7D%7D%7D%7D DOAJ] {| class="wikitable sortable" ! Journal !! Notes |- | [http://www.ac-psych.org/ Advances in Cognitive Psychology] | |- | [http://www.discourseunit.com/arcp.htm Annual Review of Critical Psychology] | |- | [http://www.apcj.org/ Applied Psychology In Criminal Justice] | |- | [http://www.athleticinsight.com/ Athletic Insight] | |- | [http://www.auseinet.com/journal The Australian e-Journal for the Advancement of Mental Health] | |- | [http://www.newcastle.edu.au/group/ajedp/ Australian Journal of Educational & Developmental Psychology] | [http://lamp.infosys.deakin.edu.au/era/?page=fordet&selfor=1701 ERA Rank B]. Papers accepted for publication become the copyright of the publisher. Green: Author pre- and post-prints can be made available. |- | [http://www.behavior-analyst-today.com/ The Behavior Analyst Today] | |- | [http://www.uic.edu/htbin/cgiwrap/bin/ojs/index.php?journal=bsi&page=index Behavior and Social Issues] | |- | [http://cpl.revues.org/ Current psychology letters] | |- | [http://www.uiowa.edu/~grpproc/crisp/crisp.html Current Research In Social Psychology] | |- | [http://goertzel.org/dynapsyc/dynacon.html Dynamical Psychology: An International, Interdisciplinary Journal of Complex Mental Processes] | |- | [http://ojs.lib.swin.edu.au/index.php/ejap E-journal of Applied Psychology] | ERA Rank C<br>Uses [[Open Journal Systems]]. Authors give journal a license to publish. The journal uses [Attribution-NonCommercial-NoDerivs 2.1 Australia cc-nd 2.1 Australia licensing]. |- | [http://www.internationaljournalofwellbeing.org/index.php/ijow/index International Journal of Wellbeing] | |- | [http://seab.envmed.rochester.edu/jaba/ Journal of Applied Behavior Analysis] | |- | [http://seab.envmed.rochester.edu/jeab/ Journal of Experimental Analysis of Behavior] | |- | [http://www.brains-minds-media.org/ Journal of New Media in Neural and Cognitive Science and Education] | |- | [http://www.psychologie-aktuell.com/index.php?id=200 Psychological Test and Assessment Modeling] | ERA Rank C<br>Articles are available at no-cost electronically<br>Licensing is unclear |- | [http://hrcak.srce.hr/psihologijske-teme?lang=en Psychological Topics] | ERA Rank: None |- | [http://www.sgcp.org.uk/coachingpsy/publications.cfm Special Group in Coaching Psychology] | |- | [http://bentham.org/open/topsyj The Open Psychology Journal] | ERA Rank: None<br>Authors who publish in Bentham OPEN Journals retain copyright to their work. Articles are licensed under the terms of the [http://creativecommons.org/licenses/by-nc/3.0/ Creative Commons Attribution non-commercial License]. [http://www.bentham.org/open/topsyj/MSandI.htm Author is required to pay an article processing fee of several hundred $US.] |} [[Category:Open journals]] [[Category:Psychology/Journals]] fjw8iuhwy48p6rdkyvivu9u9c7bn7jp 2816086 2816078 2026-06-17T06:05:39Z Jtneill 10242 See also: [[w:List of open-access journals|List of open-access journals]] (Wikipedia) 2816086 wikitext text/x-wiki ==All psychology [[journals]]== * [http://lamp.infosys.deakin.edu.au/era/?page=jfordet15&selfor=17 ERA FOR Code 17: Psychology] * [[w:List of psychology journals|List of psychology journals]] (Wikipedia) ==[[Open access]] psychology [[journals]]== This section lists open psychology journals and their rankings. * [https://www.scimagojr.com/journalrank.php?category=3201&openaccess=true scimagojr] * [https://www.doaj.org/search?source=%7B%22query%22%3A%7B%22filtered%22%3A%7B%22filter%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22term%22%3A%7B%22index.classification.exact%22%3A%22Psychology%22%7D%7D%2C%7B%22term%22%3A%7B%22_type%22%3A%22journal%22%7D%7D%5D%7D%7D%2C%22query%22%3A%7B%22match_all%22%3A%7B%7D%7D%7D%7D%7D DOAJ] {| class="wikitable sortable" ! Journal !! Notes |- | [http://www.ac-psych.org/ Advances in Cognitive Psychology] | |- | [http://www.discourseunit.com/arcp.htm Annual Review of Critical Psychology] | |- | [http://www.apcj.org/ Applied Psychology In Criminal Justice] | |- | [http://www.athleticinsight.com/ Athletic Insight] | |- | [http://www.auseinet.com/journal The Australian e-Journal for the Advancement of Mental Health] | |- | [http://www.newcastle.edu.au/group/ajedp/ Australian Journal of Educational & Developmental Psychology] | [http://lamp.infosys.deakin.edu.au/era/?page=fordet&selfor=1701 ERA Rank B]. Papers accepted for publication become the copyright of the publisher. Green: Author pre- and post-prints can be made available. |- | [http://www.behavior-analyst-today.com/ The Behavior Analyst Today] | |- | [http://www.uic.edu/htbin/cgiwrap/bin/ojs/index.php?journal=bsi&page=index Behavior and Social Issues] | |- | [http://cpl.revues.org/ Current psychology letters] | |- | [http://www.uiowa.edu/~grpproc/crisp/crisp.html Current Research In Social Psychology] | |- | [http://goertzel.org/dynapsyc/dynacon.html Dynamical Psychology: An International, Interdisciplinary Journal of Complex Mental Processes] | |- | [http://ojs.lib.swin.edu.au/index.php/ejap E-journal of Applied Psychology] | ERA Rank C<br>Uses [[Open Journal Systems]]. Authors give journal a license to publish. The journal uses [Attribution-NonCommercial-NoDerivs 2.1 Australia cc-nd 2.1 Australia licensing]. |- | [http://www.internationaljournalofwellbeing.org/index.php/ijow/index International Journal of Wellbeing] | |- | [http://seab.envmed.rochester.edu/jaba/ Journal of Applied Behavior Analysis] | |- | [http://seab.envmed.rochester.edu/jeab/ Journal of Experimental Analysis of Behavior] | |- | [http://www.brains-minds-media.org/ Journal of New Media in Neural and Cognitive Science and Education] | |- | [http://www.psychologie-aktuell.com/index.php?id=200 Psychological Test and Assessment Modeling] | ERA Rank C<br>Articles are available at no-cost electronically<br>Licensing is unclear |- | [http://hrcak.srce.hr/psihologijske-teme?lang=en Psychological Topics] | ERA Rank: None |- | [http://www.sgcp.org.uk/coachingpsy/publications.cfm Special Group in Coaching Psychology] | |- | [http://bentham.org/open/topsyj The Open Psychology Journal] | ERA Rank: None<br>Authors who publish in Bentham OPEN Journals retain copyright to their work. Articles are licensed under the terms of the [http://creativecommons.org/licenses/by-nc/3.0/ Creative Commons Attribution non-commercial License]. [http://www.bentham.org/open/topsyj/MSandI.htm Author is required to pay an article processing fee of several hundred $US.] |} See also: [[w:List of open-access journals|List of open-access journals]] (Wikipedia) [[Category:Open journals]] [[Category:Psychology/Journals]] gd5enn4dynzqdiibdh3p82pu77pi8l0 0 96387 2816067 2812286 2026-06-17T05:39:42Z Jtneill 10242 /* Marking criteria */ + (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) 2816067 wikitext text/x-wiki {{title|Book chapter — Guidelines}} <div style="text-align: center;">''Collaborative online book chapter authoring'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 09 |day = 29 |hour = 0 |event = this assessment is due }} --><!-- {{Motivation and emotion/Assessment/In development}} --> {{/Contents/}}</div> {{TOCright}} ==Overview== * Weight: 50% * Due: {{/Due}} * Tasks ** Author an online [[Motivation and emotion/Book|book chapter]] up to 4,000 words that explains key psychological theory and research about a unique, specific motivation or emotion topic ** Create the chapter by building on the plan and addressing feedback from the [[Motivation and emotion/Assessment/Topic|topic development]] exercise ** Includes a social contribution component which involves contributing to the development of other book chapters * Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] *Feedback will be provided to explain how well the chapter meets the marking criteria *Marks and feedback should be returned within 3 weeks of the due date **Marks will be available via {{Motivation and emotion/Canvas}}—keep an eye on Announcements **Written feedback will be available via the chapter's Wikiversity discussion page *Follow up if you don't understand the feedback ==Extensions and late submissions== * Extension requests require an Extension Application Form to be submitted via {{Motivation and emotion/Canvas}} with appropriate documentary evidence * Submissions are accepted up to 3 days late (-10% per day late) * If you don't submit this assessment it is unlikely that you will pass the unit ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto; vertical-align:top;" |- | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | style="vertical-align:top;" | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic. |- | style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | style="vertical-align:top;" | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Be professional—communicate effectively | style="vertical-align:top;" | Review scholarly knowledge in an open, online environment and address feedback. |- | style="vertical-align:top;" | Be professional—display initiative and drive | style="vertical-align:top;" | Produce an online book chapter about a novel motivation or emotion topic. |- | style="vertical-align:top;" | Be professional—up-to-date knowledge and skills | style="vertical-align:top;" | Utilise the most relevant psychological theory and research to address a practical question. |- | style="vertical-align:top;" | Be professional—solve problems via thinking | style="vertical-align:top;" | Use critical thinking to explain how psychological science can address real-world problems. |- | style="vertical-align:top;" | Be a global citizen—informed and balanced | style="vertical-align:top;" | Provide a balanced, critical chapter which is accessible to a lay audience. |- | style="vertical-align:top;" | Be a global citizen—communicate diversely | style="vertical-align:top;" | Collaborate with peers to communicate knowledge openly with a global audience. |- | style="vertical-align:top;" | Be a global citizen—creative use of technology | style="vertical-align:top;" | Learn how to collaborate using wiki technology. |- | style="vertical-align:top;" | Be a lifelong learner—engage in new ideas | style="vertical-align:top;" | Engage in a collaborative learning culture by incorporating feedback and suggestions. |- | style="vertical-align:top;" | Be a lifelong learner—evaluate and adopt new technology | style="vertical-align:top;" | Experience project work in a collaborative, online editing environment. |} ==Instructions== The following instructions should be used to guide the development of the book chapter. ===Theme=== * Chapters should fit the book theme which is "understanding and improving our motivational and emotional lives using psychological science" ===Audience=== * The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise. ===Wikiversity=== * Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name ===Topic=== * The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]] ===Collaboration and feedback=== * Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others) * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread<!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)-->) * Feedback is usually best placed on the chapter's wiki discussion page * Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]] ===Length (word count)=== {{Anchor|Wordcount}}{{Anchor|Word count}} * There is no minimum length * Maximum 4,000 words ** There is no additional 10% allowance ** Words beyond the maximum will not be considered for marking purposes ** Count everything from top to bottom of the editable page (in view mode, not edit mode): *** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links ** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor * If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]] ===Submission=== * Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] Book chapters will be marked against the following criteria. ===Overview (5%)=== * Provide an engaging scenario or case study * Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved (problem statement) * Establish [[/Focus questions|focus questions]] which align with the sub-title and heading structure ===Theory (20%)=== * Clearly explain the theoretical framework for understanding the topic * Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories * Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) * Clearly explain and apply the theory(ies) * Include illustrative examples, such as case studies * Demonstrate a critical perspective ===Research (25%)=== * Explain how key, peer-reviewed research findings apply to the problem * Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) * Include relevant major reviews (systematic reviews, meta-analyses etc.) * Demonstrate [[w:Critical thinking|critical thinking]]. Critically analyse key research findings, including limitations and implications. ===Integration (10%)=== * Integrate discussion of theory and review of relevant research * Use research to critically inform interpretation and application of the theory(ies) ===Conclusion (5%)=== * Emphasise the key points and take-home messages, particularly in relation to the subtitle and focus questions, with implications for personal growth and development ===Style (20%)=== * Overall ** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]] ** [[/Readability|Readable]] for a layperson interested in psychological science ** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information ** Address an international audience (i.e., avoid an overly local or national perspective) ** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles ** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment) ** Correct grammar (e.g., see [[/Writing tips|writing tips]]) * Structure ** Use a logical heading structure that aligns with the focus questions ** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings ** Use the default heading style (e.g., do not add italics and/or bold) ** Sub-headings are optional *** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings. *** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections. * Sentences ** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., "it", "they"). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., "I" and "we") and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., "you") can work well. * Paragraphs ** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs. ** Paragraphs flow logically * Use APA style (as much as reasonably possible), paying particular attention to: ** citations ** references (especially capitalisation, italicisation, and providing hyperlinked dois) ** table and figure captions ** quotes (include page numbers) * Citations ** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice. ** Maximum of 3 citations per point (i.e., avoid 4 or more citations together). * References ** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style. ** Non-academic sources are not used in references. They can be included in the external links section. ===Learning features (5%)=== * Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the "See also" and "External links" sections * Case studies ** Include 1 or more examples, scenarios, or case studies ** They can be true (if so, include citations) or fictional ** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages ** Present in a feature box and include a figure ** Consider using a "progressive case study" (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes). ** Examples of chapters which make effective use of case studies: *** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019) *** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019) *** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019) *** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019) * [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]] ** Use to highlight key information, but avoid overuse ** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option. * [[Motivation and emotion/Wikiversity/Figures|Figures]] ** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts ** Figures are accompanied by explanatory captions and be cited at least once in the main text ** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]). * [[Help:Links|Links]] ** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example: *** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is <nowiki>[[w:Emotion|emotion]]</nowiki>). It is also possible to link to a section on this same page e.g., &lt;nowiki&gt;[[#Overview|Overview]]&lt;nowiki&gt; will link to the Overview section. *** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages. ** See also *** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles *** Include source in parentheses ** External links *** Provide at least three links to high quality, relevant external resources *** Include author and/or source in parentheses ** Published academic sources belong in References * [[Motivation and emotion/Wikiversity/Tables|Tables]] ** Use accompanying tables to help organise information and communicate concepts to readers ** Tables are accompanied by explanatory APA style captions. [[Motivation and emotion/Assessment/Chapter/Tables|See example]]. * [[Help:Quiz|Quizzes]] ** Quiz questions or reflection questions encourage reader engagement ** Focus on core concepts (esp. take-home messages) rather than trivia ** Consider incorporating throughout the chapter {{anchor|Socialcontribution}} ===Social contribution (10%)=== * '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include: ** '''edits''': direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]]. ** '''comments''': feedback provided on book chapter [[Help:Talk page|talk page]]s ** '''media uploads''': create and/or upload free-to-use images to [[commons:|Wikimedia Commons]] ** '''{{Motivation and emotion/Canvas}} discussion posts''' * '''Evidence''': Provide a numbered list of social contributions on your [[Help:User page|Wikiversity user page]], with direct links to changes. To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Then summarise the edit on your user page (in a section called "Social contributions") using a numbered list and provide hyperlinks to direct evidence of the changes made. More info: [[/Summarising social contributions|summarising social contributions]]. * '''Marking''' ** Marking of social contributions will be based on: *** '''quantity''' (breadth): **** frequency: number of different chapters contributed to **** channels: range of communication channels used *** '''quality''' (depth): *** insightfulness **** practical value **** extent/thoroughness *** '''timeliness''' — there is generally: **** greater value in earlier contributions **** lesser value in "last minute" feedback ** Marks will be allocated to each clearly evidenced social contribution as follows: *** Minor <= 0.25% *** Moderate 0.50% *** Major 1.00% *** Very significant > 1.00% *** Up to 5 bonus marks may be awarded for exceptional levels of contribution ;Rubric for social contribution marking {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''Bonus marks''' | Up to 5 bonus marks are available in exceptional circumstances where wiki contributions to the book are above and beyond those required for HD-level. Such contributions could include very substantial contributions across multiple chapters. This could include extensive copyediting, regular feedback, and support on multiple chapter discussion pages. It may also involve substantial activity on the {{Motivation and emotion/Canvas}} discussion. |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | Very significant contributions are made to development of other book chapters (beyond one's target chapter). The contributor clearly embraced the collaborative nature of the online book task. This is indicated primarily by the user's edit history on Wikiversity which shows significant and regular contributions to the development of at least several chapters via discussion page comments and probably also chapter edits. Such contributions are likely to have occurred across at least half of the semester. It is also quite likely that contributions extend across more than one channel of electronically logged communication (e.g., wiki contributions and {{Motivation and emotion/Canvas}} discussion). Helping to significantly improve at least four other chapters is likely to be worth a HD. |- | style="vertical-align:top;" | '''DI (Distinction)''' | Significant contributions are made to other book chapters (beyond one's target chapter). The contributor embraced online collaboration as indicated by the user's wiki edit history. Notable contributions are made to the development of several chapters via discussion pages and chapter edits. Contributions are spread over at least a month. Contributions are likely to have extended across more than one publicly logged electronic communication channels (e.g., wiki contributions and {{Motivation and emotion/Canvas}} discussion). Helping to significantly improve at least three others chapters is likely to be worth a DI. |- | style="vertical-align:top;" | '''CR (Credit)''' | Moderate contributions to other book chapters (beyond one's target chapter). The contributor embraced some aspects of online collaboration by providing many wiki edits beyond the contributor's target chapter and/or {{Motivation and emotion/Canvas}} discussion posts. These contributions are made over a period of at least a couple of weeks and in sufficient time for other authors to incorporate the feedback into the final drafting process. As a guide, helping to significantly improve at least two other chapters is likely to be worth a CR. |- | style="vertical-align:top;" | '''P (Pass)''' | Basic contributions are made to other book chapters (beyond one's target chapter). For example, at least one other chapter in the book is significantly enhanced because of the user's contributions. This might involve some helpful comments on several occasions about at least one other book chapter — or perhaps a single, substantial proofread with several useful comments about a full draft could be sufficient for a Pass. |- | style="vertical-align:top;" | '''F (Fail)''' | Either no contributions are made or contributions were limited. A lack of collaborative effort is evident, as indicated by minimal, if any, wiki contributions beyond one's primary chapter and/or {{Motivation and emotion/Canvas}}. For example: # comments lacked detail and/or depth; # comments were not timely (e.g., were provided very late in the drafting process) |} ==Grade descriptions== This section describes typical characteristics of chapters at each grade level, based on the [[#Marking criteria|marking criteria]]. {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | A professional, near-publishable, interesting, informative, insightful, [[/Readability|readable]] explanation of relevant psychological theory and research about a well-defined, unique motivation or emotion topic. The chapter has a well-organised layout and headings, with relevant and well-captioned accompanying figures, tables, and/or figures. Excellent spelling, grammar, and APA style is used. The chapter makes effective use of wiki links to other relevant chapters and/or Wikipedia articles. Additional interactive learning features are included. Substantial social contributions are made to the development of other chapters, such as particularly useful peer-review comments on several chapter talk pages across at least half of the semester. |- | style="vertical-align:top;" | '''DI (Distinction)''' | A very good chapter, with several professional-level aspects. The chapter is informative, accurate and insightful, covering key relevant theory and research. The material is very competently handled and well-written, with minimal spelling and grammar issues. Layout is clear and effective. Good use is made of wiki links, tables, and figures. References are in very good APA style. The chapter includes additional learning features. Helpful contributions were made to some other chapters over at least a month. |- | style="vertical-align:top;" | '''CR (Credit)''' | A competent chapter with reasonably informative and insightful content which includes moderately good coverage of relevant theory and research. Some aspects of the theory or research coverage may be missing, limited, or problematic. Integration of theory and research is less assured than at higher levels. Layout and headings are reasonably useful, but could probably also be improved (e.g., by being more detailed). References are in reasonable APA style, but often several corrections are needed. Some wiki links, figures, and/or additional learning features are provided, but could have been developed further. Some helpful contributions were made over at least a couple of weeks to at least a couple of other chapters. |- | style="vertical-align:top;" | '''P (Pass)''' | The chapter provides a satisfactory, basic explanation of relevant theory and research, but lacks the depth and/or comprehensiveness that is characteristic of higher grade chapters. The chapter may struggle to clearly conceptualise the topic, organise the structure and layout, contribute to the book theme, and/or may lack depth and originality. Spelling and grammar problems are often prevalent. Citation and referencing tends to be limited in scope and quality, often with reliance on only a few (or less) high-quality peer-review references. There may relatively little meaningful use of figures or additional learning features. These chapters typically have a brief edit history (e.g., less than 2 weeks) and often read like an early draft which would benefit from more drafting to address feedback, and better proofreading. Often chapters of this standard are noticeably shorter than chapters which attract higher grades. Chapter authors often haven't sought or acted upon feedback. Some useful social contributions to at some other chapters are made, but this tends to be fairly basic and made towards the end of the drafting period. |- | style="vertical-align:top;" | '''F (Fail)''' | The chapter does not demonstrate a satisfactory grasp of key psychological theory and research which pertains to a specific, unique motivation or emotion topic. Major gaps and/or errors in content are evident, sometimes with little to no use of peer-reviewed references. These chapters typically have underdeveloped heading structures and the content is often brief or incomplete. Layout and [[/Readability|readability]] is often poor and the quality of written expression is often undermined by poor spelling and/or grammar. Sometimes plagiarism may be evident. Generally, there is a lack of sufficient effort (e.g., these chapters often have short tend to have last-minute editing histories) and have attracted little, if any, peer review. Little to no social contribution is made to the development of other book chapters. |} ==Examples== Examples of high quality motivation and emotion book chapters: * [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022) * [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016) * [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019) * [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019) Note that as of 2025, chapters no longer include multimedia presentations. For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]<!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] -->. ==Licensing== Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener. ==See also== * [[/Feature boxes/]] * [[/Figures/]] ** [[How to find free-to-use images|Find free images]] * [[/FAQ/]] * [[Motivation and emotion/Book|Previous chapters]] * [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback about book chapters]] * [[#Socialcontribution|Social contributions]] ** [[/Search for chapters to improve/]] ** [[/Summarising social contributions/]] * [[/Tables/]] * [[Motivation and emotion/Tutorials|Tutorials]] <!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --> ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] ** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]] ** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]] * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] * [[/Writing tips/]] ** [[/How to handle a lack of information/|Handling a lack of information]] ** [[/Word count|Reducing word count]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Chapter| ]] [[Category:Motivation and emotion guidelines]] m34enqcvsqb9tai2lmtcvnh1h675a3h 2816070 2816067 2026-06-17T05:43:20Z Jtneill 10242 /* See also */ Organise 2816070 wikitext text/x-wiki {{title|Book chapter — Guidelines}} <div style="text-align: center;">''Collaborative online book chapter authoring'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 09 |day = 29 |hour = 0 |event = this assessment is due }} --><!-- {{Motivation and emotion/Assessment/In development}} --> {{/Contents/}}</div> {{TOCright}} ==Overview== * Weight: 50% * Due: {{/Due}} * Tasks ** Author an online [[Motivation and emotion/Book|book chapter]] up to 4,000 words that explains key psychological theory and research about a unique, specific motivation or emotion topic ** Create the chapter by building on the plan and addressing feedback from the [[Motivation and emotion/Assessment/Topic|topic development]] exercise ** Includes a social contribution component which involves contributing to the development of other book chapters * Follow the [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] *Feedback will be provided to explain how well the chapter meets the marking criteria *Marks and feedback should be returned within 3 weeks of the due date **Marks will be available via {{Motivation and emotion/Canvas}}—keep an eye on Announcements **Written feedback will be available via the chapter's Wikiversity discussion page *Follow up if you don't understand the feedback ==Extensions and late submissions== * Extension requests require an Extension Application Form to be submitted via {{Motivation and emotion/Canvas}} with appropriate documentary evidence * Submissions are accepted up to 3 days late (-10% per day late) * If you don't submit this assessment it is unlikely that you will pass the unit ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto; vertical-align:top;" |- | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | style="vertical-align:top;" | Use the most relevant theories and peer-reviewed research to explain a specific motivation or emotion topic. |- | style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | style="vertical-align:top;" | Explain how psychological science can be applied to a specific motivation or emotion topic. Use figures, examples, and/or other interactive learning features to illustrate how this knowledge can apply to understanding human behaviour in everyday life. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- | style="vertical-align:top;" | Be professional—communicate effectively | style="vertical-align:top;" | Review scholarly knowledge in an open, online environment and address feedback. |- | style="vertical-align:top;" | Be professional—display initiative and drive | style="vertical-align:top;" | Produce an online book chapter about a novel motivation or emotion topic. |- | style="vertical-align:top;" | Be professional—up-to-date knowledge and skills | style="vertical-align:top;" | Utilise the most relevant psychological theory and research to address a practical question. |- | style="vertical-align:top;" | Be professional—solve problems via thinking | style="vertical-align:top;" | Use critical thinking to explain how psychological science can address real-world problems. |- | style="vertical-align:top;" | Be a global citizen—informed and balanced | style="vertical-align:top;" | Provide a balanced, critical chapter which is accessible to a lay audience. |- | style="vertical-align:top;" | Be a global citizen—communicate diversely | style="vertical-align:top;" | Collaborate with peers to communicate knowledge openly with a global audience. |- | style="vertical-align:top;" | Be a global citizen—creative use of technology | style="vertical-align:top;" | Learn how to collaborate using wiki technology. |- | style="vertical-align:top;" | Be a lifelong learner—engage in new ideas | style="vertical-align:top;" | Engage in a collaborative learning culture by incorporating feedback and suggestions. |- | style="vertical-align:top;" | Be a lifelong learner—evaluate and adopt new technology | style="vertical-align:top;" | Experience project work in a collaborative, online editing environment. |} ==Instructions== The following instructions should be used to guide the development of the book chapter. ===Theme=== * Chapters should fit the book theme which is "understanding and improving our motivational and emotional lives using psychological science" ===Audience=== * The target audience is a general (non-topic-expert) reader interested in personal growth and development based on knowledge in psychological science (theory and research). This is a [[w:science communication|science communication]] exercise. ===Wikiversity=== * Present the chapter as a single page on the [[Main Page|English Wikiversity]] website. A link to the chapter should appear in the [[Motivation and emotion/Book|table of contents]] along with the lead author's Wikiversity user name ===Topic=== * The title and sub-title must be approved by the [[Motivation and emotion/About/Staff|unit convener]] ===Collaboration and feedback=== * Chapters should be independently developed and written primarily by the lead author, but collaboration is strongly encouraged (e.g., by incorporating useful edits and feedback from others) * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * Lead authors are encouraged to seek feedback about the chapter during the drafting process (e.g., start a {{Motivation and emotion/Canvas}} discussion thread<!-- (use the chapter title and subtitle in the subject line and include a clickable hyperlink to the chapter in the message)-->) * Feedback is usually best placed on the chapter's wiki discussion page * Feedback on the [[Motivation and emotion/Assessment/Topic|topic development]] (chapter plan) will be provided by the [[Motivation and emotion/About/Staff|unit convener]] ===Length (word count)=== {{Anchor|Wordcount}}{{Anchor|Word count}} * There is no minimum length * Maximum 4,000 words ** There is no additional 10% allowance ** Words beyond the maximum will not be considered for marking purposes ** Count everything from top to bottom of the editable page (in view mode, not edit mode): *** Include the title, subtitle, headings, text, tables, figures, references, see also, and external links ** Use this [https://chromewebstore.google.com/detail/word-counter/cbjddaobmdfhbfgdgjocbhklpmclcboe Word Counter] (Google Chrome Extension) or paste the URL into [https://hsuper.tools/web-page-word-counter Webpage Word Counter] (it will overcount by ~100 words) or cut and paste into a word processor * If you are having difficulties complying with the maximum word count, see [[/Word count|these suggestions]] ===Submission=== * Submit the chapter URL (website address), your Wikiversity user name, and a PDF of the chapter via {{Motivation and emotion/Canvas}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] Book chapters will be marked against the following criteria. ===Overview (5%)=== * Provide an engaging scenario or case study * Easy to read and understand outline of the key concepts and explanation of practical/real-world problem to be solved (problem statement) * Establish [[/Focus questions|focus questions]] which align with the sub-title and heading structure ===Theory (20%)=== * Clearly explain the theoretical framework for understanding the topic * Select the most relevant psychological theories/models that apply to the problem. Depending on the topic, this may involve focusing on a single theory or comparing and contrasting two or more theories * Use at least the best dozen or so peer-reviewed theory references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) * Clearly explain and apply the theory(ies) * Include illustrative examples, such as case studies * Demonstrate a critical perspective ===Research (25%)=== * Explain how key, peer-reviewed research findings apply to the problem * Use at least the best dozen or so peer-reviewed research references about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) * Include relevant major reviews (systematic reviews, meta-analyses etc.) * Demonstrate [[w:Critical thinking|critical thinking]]. Critically analyse key research findings, including limitations and implications. ===Integration (10%)=== * Integrate discussion of theory and review of relevant research * Use research to critically inform interpretation and application of the theory(ies) ===Conclusion (5%)=== * Emphasise the key points and take-home messages, particularly in relation to the subtitle and focus questions, with implications for personal growth and development ===Style (20%)=== * Overall ** Present and illustrate the problem and knowledge in an interesting way, using a logical structure, clear layout, correct spelling and grammar, and [[APA style]] ** [[/Readability|Readable]] for a layperson interested in psychological science ** Address the [[#Theme|book theme]] by providing practical, academically sound, self-improvement information ** Address an international audience (i.e., avoid an overly local or national perspective) ** Use default wiki style for paragraph alignment, font colour, type, and size, and heading styles ** Use Australian spelling (e.g., hypothesise, behaviour, fulfilment) rather than American spelling (e.g., hypothesize, behavior, fulfillment) ** Correct grammar (e.g., see [[/Writing tips|writing tips]]) * Structure ** Use a logical heading structure that aligns with the focus questions ** Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] throughout, including for headings and sub-headings ** Use the default heading style (e.g., do not add italics and/or bold) ** Sub-headings are optional *** Avoid having sections with a single sub-heading — each section should contain 0 or 2+ sub-headings. *** If sub-headings are used, provides at least 1 introductory paragraph before branching into sub-sections. * Sentences ** [[w:Narration#Narrative point of view|Narrative point of view]][https://www.grammarly.com/blog/first-second-and-third-person/]: In the main text, use [[w:Narration#Third-person|3rd person perspective]] (e.g., "it", "they"). Where [[w:Aside|aside]]s are used, such as examples, case studies, and feature boxes, [[w:First-person narrative|1st person perspective]] (e.g., "I" and "we") and/or [[w:Narration#Second-person|2nd person perspective]] (e.g., "you") can work well. * Paragraphs ** A well-constructed paragraph is generally 3 to 5 sentences (opening sentence, body sentences, and a concluding/linking sentence). Avoid one-sentence paragraphs and overly long paragraphs. ** Paragraphs flow logically * Use APA style (as much as reasonably possible), paying particular attention to: ** citations ** references (especially capitalisation, italicisation, and providing hyperlinked dois) ** table and figure captions ** quotes (include page numbers) * Citations ** Claims need citations using APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style throughout the chapter — don't mix and match. For most psychology students, APA style will be the choice. ** Maximum of 3 citations per point (i.e., avoid 4 or more citations together). * References ** List all cited academic references in APA style or [[w:Wikipedia:Citing_sources|wiki citation style]]. Only use one style. ** Non-academic sources are not used in references. They can be included in the external links section. ===Learning features (5%)=== * Embed interactive learning features such as scenarios/case studies/examples, feature boxes, figures, quizzes, links to relevant Wikipedia and Wikipedia pages, as well as links to key resources via the "See also" and "External links" sections * Case studies ** Include 1 or more examples, scenarios, or case studies ** They can be true (if so, include citations) or fictional ** Use these examples to enhance understanding of theory, research, focus questions, and/or take-home messages ** Present in a feature box and include a figure ** Consider using a "progressive case study" (i.e., a case study presented in separate parts which describe, for example, the problem, attempt at change, and resolution/outcomes). ** Examples of chapters which make effective use of case studies: *** [[Motivation and emotion/Book/2019/Emotional abuse|emotional abuse]] (2019) *** [[Motivation and emotion/Book/2019/Food and fear|food and fear]] (2019) *** [[Motivation and emotion/Book/2019/Opioid system and human emotion|opioid system and human emotion]] (2019) *** [[Motivation and emotion/Book/2019/Social support and emotion|social support and emotion]] (2019) * [[Motivation and emotion/Wikiversity/Feature box|Feature boxes]] ** Use to highlight key information, but avoid overuse ** There are various ways of creating coloured boxes, but the [[Template:RoundBoxTop|RoundBox]] template is a good option. * [[Motivation and emotion/Wikiversity/Figures|Figures]] ** Include relevant, accompanying figures (e.g., photos, drawings, diagrams) to facilitate readers' understanding of the concepts ** Figures are accompanied by explanatory captions and be cited at least once in the main text ** For more information, see [[Motivation and emotion/Assessment/Chapter/Figures|How to use figures]]). * [[Help:Links|Links]] ** In-text (embedded) links: Key words and concepts are [[Making links|linked]] to Wikipedia articles and/or related book chapters. Provide in-text wiki links the ''first time'' that key concepts are mentioned. For example: *** [[w:Emotion|emotion]] involves physiological, subjective feeling, motivational, and socially expressive aspects. The syntax for creating this link is <nowiki>[[w:Emotion|emotion]]</nowiki>). It is also possible to link to a section on this same page e.g., &lt;nowiki&gt;[[#Overview|Overview]]&lt;nowiki&gt; will link to the Overview section. *** [[Motivation and emotion/Book/2021/Fitspiration and body image|This chapter]] provides an excellent example of embedded links to Wikiversity pages. ** See also *** Provide interwiki links to key related Wikiversity book chapters and/or Wikipedia articles *** Include source in parentheses ** External links *** Provide at least three links to high quality, relevant external resources *** Include author and/or source in parentheses ** Published academic sources belong in References * [[Motivation and emotion/Wikiversity/Tables|Tables]] ** Use accompanying tables to help organise information and communicate concepts to readers ** Tables are accompanied by explanatory APA style captions. [[Motivation and emotion/Assessment/Chapter/Tables|See example]]. * [[Help:Quiz|Quizzes]] ** Quiz questions or reflection questions encourage reader engagement ** Focus on core concepts (esp. take-home messages) rather than trivia ** Consider incorporating throughout the chapter {{anchor|Socialcontribution}} ===Social contribution (10%)=== * '''Actions''': Logged contributions which enhance the quality of other book chapters. Useful actions include: ** '''edits''': direct edits which improve past or current chapters (e.g., fix errors, enhance clarity) or flag potential improvements by adding [[Template:Clarification templates|clarification templates]]. [[/Search for chapters to improve|Search for chapters to improve]]. ** '''comments''': feedback provided on book chapter [[Help:Talk page|talk page]]s ** '''media uploads''': create and/or upload free-to-use images to [[commons:|Wikimedia Commons]] ** '''{{Motivation and emotion/Canvas}} discussion posts''' * '''Evidence''': Provide a numbered list of social contributions on your [[Help:User page|Wikiversity user page]], with direct links to changes. To receive credit, contributions must be publicly logged (i.e., log in to Wikiversity so that the edit is recorded with your user name and time-stamp). Then summarise the edit on your user page (in a section called "Social contributions") using a numbered list and provide hyperlinks to direct evidence of the changes made. More info: [[/Summarising social contributions|summarising social contributions]]. * '''Marking''' ** Marking of social contributions will be based on: *** '''quantity''' (breadth): **** frequency: number of different chapters contributed to **** channels: range of communication channels used *** '''quality''' (depth): *** insightfulness **** practical value **** extent/thoroughness *** '''timeliness''' — there is generally: **** greater value in earlier contributions **** lesser value in "last minute" feedback ** Marks will be allocated to each clearly evidenced social contribution as follows: *** Minor <= 0.25% *** Moderate 0.50% *** Major 1.00% *** Very significant > 1.00% *** Up to 5 bonus marks may be awarded for exceptional levels of contribution ;Rubric for social contribution marking {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''Bonus marks''' | Up to 5 bonus marks are available in exceptional circumstances where wiki contributions to the book are above and beyond those required for HD-level. Such contributions could include very substantial contributions across multiple chapters. This could include extensive copyediting, regular feedback, and support on multiple chapter discussion pages. It may also involve substantial activity on the {{Motivation and emotion/Canvas}} discussion. |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | Very significant contributions are made to development of other book chapters (beyond one's target chapter). The contributor clearly embraced the collaborative nature of the online book task. This is indicated primarily by the user's edit history on Wikiversity which shows significant and regular contributions to the development of at least several chapters via discussion page comments and probably also chapter edits. Such contributions are likely to have occurred across at least half of the semester. It is also quite likely that contributions extend across more than one channel of electronically logged communication (e.g., wiki contributions and {{Motivation and emotion/Canvas}} discussion). Helping to significantly improve at least four other chapters is likely to be worth a HD. |- | style="vertical-align:top;" | '''DI (Distinction)''' | Significant contributions are made to other book chapters (beyond one's target chapter). The contributor embraced online collaboration as indicated by the user's wiki edit history. Notable contributions are made to the development of several chapters via discussion pages and chapter edits. Contributions are spread over at least a month. Contributions are likely to have extended across more than one publicly logged electronic communication channels (e.g., wiki contributions and {{Motivation and emotion/Canvas}} discussion). Helping to significantly improve at least three others chapters is likely to be worth a DI. |- | style="vertical-align:top;" | '''CR (Credit)''' | Moderate contributions to other book chapters (beyond one's target chapter). The contributor embraced some aspects of online collaboration by providing many wiki edits beyond the contributor's target chapter and/or {{Motivation and emotion/Canvas}} discussion posts. These contributions are made over a period of at least a couple of weeks and in sufficient time for other authors to incorporate the feedback into the final drafting process. As a guide, helping to significantly improve at least two other chapters is likely to be worth a CR. |- | style="vertical-align:top;" | '''P (Pass)''' | Basic contributions are made to other book chapters (beyond one's target chapter). For example, at least one other chapter in the book is significantly enhanced because of the user's contributions. This might involve some helpful comments on several occasions about at least one other book chapter — or perhaps a single, substantial proofread with several useful comments about a full draft could be sufficient for a Pass. |- | style="vertical-align:top;" | '''F (Fail)''' | Either no contributions are made or contributions were limited. A lack of collaborative effort is evident, as indicated by minimal, if any, wiki contributions beyond one's primary chapter and/or {{Motivation and emotion/Canvas}}. For example: # comments lacked detail and/or depth; # comments were not timely (e.g., were provided very late in the drafting process) |} ==Grade descriptions== This section describes typical characteristics of chapters at each grade level, based on the [[#Marking criteria|marking criteria]]. {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | A professional, near-publishable, interesting, informative, insightful, [[/Readability|readable]] explanation of relevant psychological theory and research about a well-defined, unique motivation or emotion topic. The chapter has a well-organised layout and headings, with relevant and well-captioned accompanying figures, tables, and/or figures. Excellent spelling, grammar, and APA style is used. The chapter makes effective use of wiki links to other relevant chapters and/or Wikipedia articles. Additional interactive learning features are included. Substantial social contributions are made to the development of other chapters, such as particularly useful peer-review comments on several chapter talk pages across at least half of the semester. |- | style="vertical-align:top;" | '''DI (Distinction)''' | A very good chapter, with several professional-level aspects. The chapter is informative, accurate and insightful, covering key relevant theory and research. The material is very competently handled and well-written, with minimal spelling and grammar issues. Layout is clear and effective. Good use is made of wiki links, tables, and figures. References are in very good APA style. The chapter includes additional learning features. Helpful contributions were made to some other chapters over at least a month. |- | style="vertical-align:top;" | '''CR (Credit)''' | A competent chapter with reasonably informative and insightful content which includes moderately good coverage of relevant theory and research. Some aspects of the theory or research coverage may be missing, limited, or problematic. Integration of theory and research is less assured than at higher levels. Layout and headings are reasonably useful, but could probably also be improved (e.g., by being more detailed). References are in reasonable APA style, but often several corrections are needed. Some wiki links, figures, and/or additional learning features are provided, but could have been developed further. Some helpful contributions were made over at least a couple of weeks to at least a couple of other chapters. |- | style="vertical-align:top;" | '''P (Pass)''' | The chapter provides a satisfactory, basic explanation of relevant theory and research, but lacks the depth and/or comprehensiveness that is characteristic of higher grade chapters. The chapter may struggle to clearly conceptualise the topic, organise the structure and layout, contribute to the book theme, and/or may lack depth and originality. Spelling and grammar problems are often prevalent. Citation and referencing tends to be limited in scope and quality, often with reliance on only a few (or less) high-quality peer-review references. There may relatively little meaningful use of figures or additional learning features. These chapters typically have a brief edit history (e.g., less than 2 weeks) and often read like an early draft which would benefit from more drafting to address feedback, and better proofreading. Often chapters of this standard are noticeably shorter than chapters which attract higher grades. Chapter authors often haven't sought or acted upon feedback. Some useful social contributions to at some other chapters are made, but this tends to be fairly basic and made towards the end of the drafting period. |- | style="vertical-align:top;" | '''F (Fail)''' | The chapter does not demonstrate a satisfactory grasp of key psychological theory and research which pertains to a specific, unique motivation or emotion topic. Major gaps and/or errors in content are evident, sometimes with little to no use of peer-reviewed references. These chapters typically have underdeveloped heading structures and the content is often brief or incomplete. Layout and [[/Readability|readability]] is often poor and the quality of written expression is often undermined by poor spelling and/or grammar. Sometimes plagiarism may be evident. Generally, there is a lack of sufficient effort (e.g., these chapters often have short tend to have last-minute editing histories) and have attracted little, if any, peer review. Little to no social contribution is made to the development of other book chapters. |} ==Examples== Examples of high quality motivation and emotion book chapters: * [[Motivation and emotion/Book/2022/Disappointment|Disappointment]]: What is disappointment, what causes it, and how can it be managed? (2022) * [[Motivation and emotion/Book/2016/Illicit drug taking at music festivals|Illicit drug taking at music festivals]]: What motivates young people to take illicit drugs at music festivals? (2016) * [[Motivation and emotion/Book/2019/Organisational change motivation|Organisational change motivation]]: How can leaders build a culture of agility, adaptability, and resilience to deal with a constantly changing workplace? (2019) * [[Motivation and emotion/Book/2019/Phobias|Phobias]]: What are phobias and how can they be dealt with? (2019) Note that as of 2025, chapters no longer include multimedia presentations. For more examples, see the {{Motivation and emotion/Book/High}}s in the [[Motivation and emotion/Book|lists of previous book chapters]]<!-- and the [[:Category:Motivation and emotion/Book/2022/Top|top chapters of 2022]] -->. ==Licensing== Contributions to Wikiversity are made under a [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 Share-alike] (CC-BY-SA 4.0) license which is irrevocable. This license gives permission for others to edit and re-use, with appropriate acknowledgement. For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under this license, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment options]] with the unit convener. ==See also== * [[/FAQ/]] * [[Motivation and emotion/Book|Previous chapters]] * Marking and feedback ** [[Motivation and emotion/Assessment/Chapter/Feedback|General feedback]] ** [[Template:MEBF|Feedback template]] * [[#Socialcontribution|Social contributions]] ** [[/Search for chapters to improve/]] ** [[/Summarising social contributions/]] * [[Motivation and emotion/Tutorials|Tutorials]] <!-- ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] --> ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] ** [[Motivation and emotion/Tutorials/Functionalist theory and self-tracking#Google Scholar|Tutorial 05: Google Scholar]] ** [[Motivation and emotion/Tutorials/Measuring emotion#Topic development feedback|Tutorial 08: Topic development feedback]] * Wikiversity ** [[/Feature boxes/]] ** [[/Figures/]] ** [[How to find free-to-use images|Find free images]] ** [[/Tables/]] * [[/Writing tips/]] ** [[/How to handle a lack of information/|Handling a lack of information]] ** [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] ** [[/Word count|Reducing word count]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Chapter| ]] [[Category:Motivation and emotion guidelines]] elfd4p06yc9dpqe3d0a2ie2b7gdo9qe 0 99140 2816050 2191136 2026-06-17T00:28:57Z Jtneill 10242 added [[Category:Motivation and emotion/Book/Health]] using [[Help:Gadget-HotCat|HotCat]] 2816050 wikitext text/x-wiki {{title|Stress and health}} {{MECR|http://screenr.com/wsc}} {{MET}} __TOC__ ==What is stress?== [[Image:Worried People 2.jpg|thumb|300px|right|Stress is experienced by almost all people and can be both beneficial and detrimental.]] Imagine yourself in the following situation: It is the morning of an important exam, and you are feeling worried because you think that you aren’t quite prepared. You get into your car, but have trouble starting the engine. You crank the motor again and again without any luck. Finally, the motor roars into life and you are on your way to university. You are already running a little late, but think that you can still make it to your exam on time if you drive a little faster than you probably should – until you turn onto the main road to the university to find that traffic is at a standstill. Now, think of the different physical sensations and emotions you would be feeling in the aforementioned situation. You would probably feeling anxious and worried, your heart would be racing, you would be sweating, your stomach would be turning and you would suddenly feel highly alert. These are all symptoms of the stress response. Life is inherently stressful and almost every person experiences stress at some stage in their life. [[w:Stress (biology)|Stress]] is a non-specific emotional and physiological response to demands in the environment, both positive and negative (Baum & Posluszny, 1999). In an evolutionary context, stress is an extremely important defensive reaction, activating the ‘fight or flight’ response – a process by which the body prepares to either face or run away from danger. The stress response is a multifaceted process that occurs when individuals are faced with stressors (stimuli that elicit stress) and involves both psychological and physiological elements. ==The stress response== The stress response is an automatic physical and emotional response to stimuli that is perceived to be a threat to safety. Unconsciously, the cerebral cortex alerts the [[w:hypothalamus|hypothalamus]] and the pituitary gland,which send signals to the adrenal cortex via the peripheral nervous system to release epinephrine and norepinephrine into the blood stream. Simultaneously, the pituitary gland sends signals to the adrenal cortex via adrenocorticotropic hormone (ACTH) that is released into the blood. ACTH stimulates the adrenal glands to secrete mineralocorticoids and glucocorticoids (Marieb & Hoehn, 2007; Myers, 2007). These hormones affect a number of organs and systems in the body to produce the following symptoms: * Increased heart rate * Increased blood pressure * Increase blood-sugar concentration * Widening of the arteries * Diversion of blood away from stomach * Increased blood flow to skeletal muscles * Increased muscle tension and strength (Myers; Marieb & Hoehn) The stress response also induces an emotional and cognitive reaction from the individual. Many emotions are elicited during the stress response such as anger, fear, anxiety or depressed mood (Baum & Posluszny, 1999). Cognitive function is also enhanced during the stress response, as mental alertness is heightened (Myers, 2007). All of these elements prepare the organism to either confront or run from the perceived danger. ==Theories of Stress== ===General Adaptation Syndrome (Selye)=== [[Image:General Adaptation Syndrome.jpg|thumb|400px|right|''Figure 1''. Diagram of the three stages of the General Adapation Syndrome (GAS)]] In 1950, Hans Selye published his theory explaining the mechanisms of stress – General Adaptation Syndrome (GAS) - in the British Medical Journal. This theory is still highly influential today in medical research. According to GAS, the response that every organism shows to stress is the same, regardless of the organism or the stressor that induces the reaction (Selye, 1950). In the GAS model, there are three stages to the stress response: the alarm reaction, resistance and exhaustion (see Figure 1). # The Alarm Reaction – In this stage, the stressor is identified by the organism and the stress response rapidly occurs. Initially, the ability to resist the stressor is diminished, due to the physiological effects caused by the activation of the sympathetic nervous system (such as increased heart rate and diversion of blood away from organs such as the stomach to the skeletal muscles) (Myers, 2007). However, the body quickly adjusts and moves into stage two, resistance (Selye, 1950). # Resistance – During this stage, the body deals with the stressor by increasing heart rate and blood pressure, increasing energy available to the skeletal muscles and heightening alertness temporarily (Marieb & Hoehn, 2007). This response is only designed to be sustained for a short period of time, after which stage 3 occurs (Selye, 1950). # Exhaustion – In stage 3 of GAS, if the presence of the stressor is sustained, the body’s resources become depleted and the body can no longer sustain an appropriate level of resistance to the stressor. The ability of the body to deal with the stressor becomes lower than it was prior to the stress response, which can leave the organism vulnerable to injury or disease (Myers, 2007; Selye, 1950). Whilst this model is widely accepted by the medical community as an appropriate conceptualisation of the stress response, Selye did not make any firm distinction between the different type of stressors and did not take into account individual differences in the way people perceive stressors (Lazarus, 1993). In this model, both psychological and physiological stressors are treated in the same way; that is, a stressor is defined as anything that is potentially damaging to tissue (Lazarus, 1993). However, in reality, individual perceptions on what is a threat and what is harmful vary largely between different people (McCraty & Tomasino, 2006). In reaction to this, research moved in the direction of explaining the psychological (particularly, the emotional and cognitive) component of the stress response. ===Cognitive Appraisal Theory (Lazarus)=== [[Image:Studying.jpg|260px|right|thumb|Studying for an exam is a stressor that is often viewed as a challenge, rather than a threat.]] The cognitive appraisal theory looks to address some of the deficits of the GAS theory and centres around the concept that it is the organism's perception and appraisal of the stressor that is the most important factor in the intensity of the stress response. This theory suggests that the stress response can be mediated by the individuals appraisal of the stressor as either a ''threat'' or a ''challenge'' (Larazus, 1990). Depending on whether the stressor is considered a threat to safety or positive challenge, the body’s reaction will differ. If the stressor is appraised to be a challenge, the stress response will be diminished and the experience will be more positive than if the stressor is perceived as a threat (which would induce the ‘typical’ stress response, defined by GAS) (Lazarus, 1993). Lazarus (1990) argues that the stress response is just one part of a larger process that is driven by cognition. This theory has been received favourably given that it takes into account the large psychological component of stress and how individual experience plays a role in the perception of stressors (Ursin & Eriksen, 2001). This theory also have implications for the management of stress, which is discussed later in this chapter. ==Relationship between Stress and Health== [[Image:Stress 2.gif|thumb|400px|left|''Figure 2''. Chronic stress has a variety of effects on the body]] Stress is one of the most important factors that mediates the relationship between [[w:Health|health]] and [[w:Behavior|behaviour]] (Baum & Posluszny, 1999). It is widely regarded as one of the most important factors in illness (Hanson & Chen, 2010) and is fast becoming a major public health problem (Gehling, Aubert, Padlina, Martin-Diener & Somaini, 2001). Acute stress can be beneficial for an organism, triggering the ‘fight or flight’ response and thereby increasing strength, stamina and attention for a short period of time (Ekman & Arnetz, 2006). However, chronic stress is disruptive to a multitude of psychological and physiological processes. Over time, if the stress response system is activated continuously, the allostatic load caused by this can put strain on the body (Hanson & Chen, 2010). The ‘exhaustion’ stage of Selye’s GAS theory states that prolonging and sustaining the physiological and psychological reactions of stress long-term has the potential to weaken the body’s immune responses through exhaustion, thereby predisposing the body to disease (Baum & Posluszny, 1999; Selye, 1950). In the long-term, the constant presence of stress hormones within the body results in a range of effects on the organs that create homeostatic imbalance. Mineralocorticoids cause the retention of water and sodium in the body by the kidneys, which increases blood volume and blood pressure. Glucocorticoids increase the amount of glucose in the blood by suppressing insulin and increasing the conversion of fat to glucose (Marieb & Hoehn, 2007). Both of these processes put strain on the cardiovascular and immune systems, as both of these systems use the blood to transport oxygen, antibodies, immune cells and nutrients to cells as well as transporting waste and infection away from cells. The psychological impacts of stress also play a role in health outcomes. Research has shown that individuals who are under chronic stress are more likely to engage in unhealthy behaviours such as excessive drinking, drug misuse, smoking, overeating and sedentary behaviour in order to manage and relieve the psychological discomfort of stress (Krueger & Chang, 2008). Increased alcohol and drug misuse has been strongly linked to stress in the research literature. The affect regulation model suggests that when stress increases, individuals 'self-medicate' with alcohol and drugs in order to reduce the feelings of distress and discomfort that come with stress (Grzywacz & Almeida, 2008). In their study into alcohol use, negative mood and stress, Grzywacz and Almeida found that the probability that a person would engage in binge drinking behaviours was largely increased on days when individuals experienced stress. Smokers also tend to report increasing their number of cigarettes they smoke during periods of high stress (Söderpalm & Söderpalm, 2006). Engaging in unhealthy behaviours long-term can result in a number of poor health outcomes such as cancer, obesity, health complications related to drug dependence or misuse and death (Krueger & Chang, 2008). Chronic stress also leads to more instances of self-reported poor health and has been shown to be a large contributor to psychosomatic illness or pain (Ursin & Eriksen, 2007). Numerous studies have found that individuals who suffer from chronic stress report significantly poorer health than individuals who do not suffer chronic stress (Holmgren, Dahlin-Ivanoff, Björnlund & Hensing, 2009; Yu, Chiu, Lin, Wang & Chen, 2007). A study of Swiss participants found that not only did those who reported suffering chronic stress report more health complaints, the severity of the symptoms that they reported was significantly more likely to fall within a clinical range (Gehling, Aubert, Padlina, Martin-Diener & Somaini, 2001). Stress also appears to play a large role in the onset of psychosomatic illness, pain and medically unexplained health conditions. Sufferers of medically unexplained syndromes (such as chronic fatigue syndrome) often report that a stressful event preceded the onset of their condition (Rubin & Wessely, 2006). ==Physiological Impacts of Stress== ===Cardiovascular=== According to figures released by the Australian Bureau of Statistics, cardiovascular disease was the number one cause of death for both males and females in Australia during 2008 (ABS, 2010). [[w:Cardiovascular disease|Cardiovascular disease]] (CVD) is a blanket term used to describe a range of disorders that affect different components of the cardiovascular system including conditions such as myocardial infarction (heart attack), stroke, hypertension (high blood pressure), arrhythmia, aneurysm, thrombosis and oedema (Marieb & Hoehn, 2007). The associations between emotional states (such as stress) and cardiovascular disease have been well established over the last 20 years and have been shown to influence the development of cardiovascular pathology independent of other risk factors like smoking, weight and diet (Grippo & Johnson, 2009). So unsurprisingly, understanding the effects of stress on cardiovascular disease is becoming an important priority for medical science. [[Image:Heart.jpg|thumb|200px|left|The heart can be damaged by chronic stress.]] The relationship between stress (both acute and chronic) and CVD is well established, particularly in relation to myocardial infarction. Schnenck-Gustatsson (2009) found that stress was one of the most important risk factors in the development of CVD in women. In fact, stress is the primary cause of one type of myocardial infarction, called takotsubo cardiomyopathy, which only affects post-menopausal women (Ueyama, Kasamatsu, Hano, Tsuruo & Ishikura, 2008). This particular type of myocardial infarction was identified after an earthquake occurred in Japan, and the number of women suffering heart attacks increases dramatically (Ueyama, Kasamatsu, Hano, Tsuruo & Ishikura, 2008). As well as major life events, such natural disasters or periods of change and uncertainty, personality can influence stress, and it has been investigated in relation to CVD. During the 1980’s, type A personality was linked with CVD. Individuals with type A personalities tend to experience more stress due their propensity to be hostile and aggressive (Friedman & Booth-Kewley, 1987). However, research into type A personality and CVD was mixed and inconclusive, and the focus in recent years has shifted to other personality types that may be related to stress, such as type D 'depressive' personality (Pozuelo, Zhang, Franco, Tesar, Penn & Jiang, 2009; Denollot, 1998). Given that a large portion of the changes that occur during the stress response are targeted at the components of the cardiovascular system such as the veins and arteries, it is unsurprising that over time, damage can also occur to these components of the cardiovascular system. High blood pressure (which is one of the primary symptoms of stress) when experienced over the long term, can weaken the walls of blood vessels and cause aneurysms to develop and rupture (Torii, Oshima, Kobayashi, Takagi & Tezduyar, 2007). Links have also been found between the presence of severe, acute stressors and the development of heart arrhythmias (Grippo & Johnson, 2009). One theme that appears from the research literature into CVD and stress is that stress is an independent risk factor (given the pressure that the stress response directly puts on the cardiovascular system), but it also exacerbates other risk factors such as diet, smoking, levels of physical activity and weight. As mentioned previously, stress impacts the likelihood of a person engaging in unhealthy behaviours (Krueger & Chang, 2008). Learning to manage stress is one of the treatment strategies taught to CVD patients to help them manage their condition and reduce the likelihood of future illness (Grippo & Johnson, 2009). {| cellpadding="10" cellspacing="5" style="float: right; width: 40%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 40%; background-color:LightGreen; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 10px;" | {{center top}} {{big2|'''Case Study: Part One'''}} {{center bottom}} [[Image:Migraine.jpg|100px|left]] Jayne is a young woman in her early 20's who suffers from chronic stress. She reports feeling stressed about several aspects of her life, including her work-life balance, her university studies as well as conflicts that have occurred between herself and some friends. Over the past 12 months, she has also noticed that her health has been suffering. Jayne has had a number of illnesses, including some recurring problems. She has had four bouts of gastroenteritis, two ear infections, several bouts of common cold as well as one bout of 'flu and a severe case of whooping cough (even though she had received vaccinations for both). She reports being 'constantly sick' and 'lacking energy'. Jayne feels as though her poor health is exacerbating her stress and is desperate to get her health back under control. |} ===Infectious Diseases=== The situation experienced by the young woman in the case study, Jayne, is not uncommon for people who suffer chronic stress. Stress has a clear impact on the [[w:Immune system|immune system]] and the wear-and-tear caused by stress can leave the body unable to mount a defense against incoming pathogens (Godbout & Glaser, 2006; Baum & Posluszny, 1999). Infectious diseases such as upper-respiratory infections, the 'flu, strains of herpes virus and hepatitis C have all been shown to spread faster in individuals experiencing chronic stress (Cohen, Janicki-Deverts & Miller, 2007). The presence of chronic stress can increase the duration and severity of infectious diseases (Cohen, Janicki-Deverts & Miller, 2007). It has also been shown to help provide an environment within the body that promotes the activation and duplication of viruses and bacteria and is also being investigated as a potential cofactor in tumour growth and development (Godbout & Glaser, 2006). One of the hormones that is released during the stress response, glucocorticoid, has been implicated as the potential primary cause of immuno-suppression (Godbout & Glaser,). Once the immune system has been suppressed, viruses and bacteria that cause infectious diseases are more easily able to multiple and create infection within the body (Baum & Posluszny, 1999). The emotional state that stress induces is also implicated in the increased propensity for those suffering chronic stress to also develop infectious diseases more rapidly (Clougherty & Kubzansky, 2009). The emotions associated with stress, such as depression and anxiety, has been shown to reduce the activity of key immune system cells such as natural killer cells and T- and B-cells (Godbout & Glaser, 2006). The suppressed state of the immune system that chronic stress induces has large implications for the management and prevention of infectious diseases. Wound healing and surgical recovery has been shown to be greatly impacted if patients are in stressful environments. Godbout and Glaser (2006) found that wound healing was far slower under stressful conditions and the potential for the wound becoming infected was largely increased. The prevention of infectious diseases can be impaired by presence of chronic stress. When reading the case study, you may have asked yourself 'how it is that Jayne could have caught the flu and whooping cough, given that she had received vaccinations against those two viruses?'. Research shows that if Jayne received these inoculations whilst she was suffering chronic stress, they may not have worked. The antibody development process that occurs after being vaccinated can be impaired if the vaccine is administered whilst the recipient is under stress (Godbout & Glaser, 2006). B-cells play an essential role in the development of antibodies, and when the body is under psychological stress, the function of these cells is greatly reduced (Clougherty & Kubzansky, 2009). ==Psychological Impacts of Stress== ===Depression=== The stress response elicits a wide range of emotions, however, it is mainly responsible for producing a state of negative affect (Krueger & Chang, 2008). Stress is considered to be one of the primary risk factors for the ontset of [[w:Major depressive disorder|major depressive disorder]] and episodes of sub-clinical depression (Husky, Mazure, Maciejewski & Swendsen, 2008). Husky et al. found that stress has a large influence on the development of an initial episode of depression, however tends to play a lesser role in every subsequent episode of depression. This finding is consistent with much of the research literature on the role of stress in depression. One study found that in the three to six months preceding the onset of depressive symptoms, 50 to 80% of depressed individuals reported experiencing at least one stressful life event (Cohen, Janicki-Deverts & Miller, 2007). The genetic predisposition to develop depression also appears to play a role in how individuals react to stressors. A recent study found that those with a family history of depression tended to report heightened negative affect in reaction to life stressors (Wichers et al., 2007). There appears to be an interplay between genetics, cognitive and environmental factors in the relationship between stress and depression. No one type of stressor appears to have a greater influence on the development of depression than any other. In a study into the effects of work-place stress on mental health, Siegrist (2008) found that work-related stress was associated with a significantly increased risk of developing depression. Childbirth is also a highly stressful life event, and post-natal depression can develop during this period of high stress and rapid change. Interestingly, the stress of having a new baby appears to influence the development of depressive symptoms equally in both mothers and fathers (Gao, Chan & Mao, 2009). However, Gao et al. (2009) found that mothers tended to be able to cope with the stress better, due to greater social support. The ability to cope with the stress appeared to mediate the level of depression that the mothers in Gao’s study suffered. Research, like the Gao et al. study, suggest that one of the factors that may contribute to the high prevalence of depressive disorders amongst adolescents and young adults may be their inability to cope with stress effectively. Sun, Tao, Hao and Wan (2010) found in a study of Chinese adolescents that the ability to cope with stressful life events reduced the likelihood of developing depression. ===Post-traumatic Stress Disorder=== [[Image:Post-traumatic stress disorder world map - DALY - WHO2004.svg|right|thumb|300px|This map shows the countries with the highest prevalence of PTSD in red.]] Post-traumatic stress disorder (PTSD) is arguably the one of the worst health outcomes that comes from exposure to stress. PTSD is a debilitating psychiatric condition that is triggered by a period of highly traumatic and stressful experiences that involved actual or threatened serious injury or death or the actual or threatened danger to personal integrity, either of the individual or others (Bisson, 2007). PTSD is most prevalent among males, and there is a high prevalence amongst veterans of war (Bisson, 2007; Stein, 2002). Following the stressful event, sufferers of PTSD experience severe and debilitating stress reactions whether or not they are in the presence of a stimuli that may remind them of the traumatic event (Yule, 1999). The stress reactions that sufferers experience when they experience flashbacks are linked with the development of depression and suicide attempts are common amongst those with PTSD who are of adolescent age and over (Yule, 1999). The neurological damage that is associated with long-term chronic stress (such as decreased hippocampus size, due to neuron atrophy) can be found in sufferers of PTSD (Stein, 2002). Even though greater than 50% of adults have experienced at least one traumatic event during their life time, only approximately 10% of these individuals suffer from PTSD (Bisson, 2007). Exactly what makes one individual more susceptible to PTSD than another is still an area of debate in psychology. One model suggests that suffering a psychological disorder earlier in life increases the chance of developing PTSD after being exposed to a traumatising stressor (Cohen, Matar, Richter-Levin & Zohar, 2006). Debate about what biological mechanisms drive PTSD is also still up for debate, although dysfunction of the same hypothalamic-pituitary-adrenal axis that is activated in the normal stress response is returning promising results in an animal model (Cohen et al., 2006). ==Preventative measures== {| cellpadding="10" cellspacing="5" style="float: left; width: 40%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 40%; background-color:LightGreen; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 10px;" | {{center top}} {{big2|'''Case Study: Part Two'''}} {{center bottom}} [[Image:Migraine.jpg|100px|left]] Jayne decided to seek help from a psychologist in order to get her stress levels under control. She was taught strategies to help her manage her stress and began exercising and mediating in order to keep stress levels under control. Jayne also stopped engaging in unhealthy behaviours, like sedentary behaviour, when she began to feel stressed. As her stress level declined over the next six months, her health improved. She was sick less frequently and began to feel more energetic. |} Given the wide variety of health problems that are associated with stress, understanding how to manage stress in order to improve health is a high priority, particularly in developed nations (Gehring, Aubert, Padlina, Martin-Diener & Somaini, 2001). Thankfully, research is beginning to reveal that the effect of stress on health can be mediated by an individual's ability to cope with the stressor (Wang, Keown, Patten, Williams, Currie, Beck, Maxwell and El-Guebaly, 2009). According to the cognitive appraisal theory, it is the perception of the stressor that has the greatest impact on the level of stress produced (Lazarus, 1990), so learning strategies to help change the way an individual perceives everyday stressors (like the traffic jam from the start of the chapter) has a large impact on the levels of stress an individual experiences (Yu, Chiu, Lin, Wang & Chen, 2007). Like Jayne, many individuals use physical exercise as an effective method of reducing stress, although it can be difficult to get people who are stressed to engage in physical activity as sedentary behaviour can be highly rewarding over the short-term (Siegrist, 2008; Baum & Posluszny, 1999). Mediation has also been shown to be another effective method of stress reduction. In their study into CVD in an African-American population, Schneider and his colleagues found the use of transcendental mediation not only reduced stress, but in doing so, it significantly reduced the risk of CVD (Schneider, Alexander, Salerno, Rainforth & Nidich, 2005). Reducing stress can help improve self-reported health, as in the case of Jayne. Given the fact that the body is not designed to deal with chronic stress (Grippo & Johnson, 2009), making sure stress levels are managed is fast become an accepted method of controlling one of today's the largest public health issues. ==Summary== Stress is an often unwelcome but common part of our lives and is widely recognised as one of the primary risk factors in health and illness (Hanson & Chen, 2010). This chapter has discussed the relationship between stress and health, looked in depth at four health conditions in which stress is a primary risk factor and discussed some of the preventative methods that can be used to reduce stress and thereby improve health. The two most prominent theories of stress - GAS and the Cognitive Appraisal theory - focus on different aspects of the stress response (the physiological and the psychological), but have both been shown to be quite useful in understanding the multifaceted nature of stress. The physiological features of the stress response impact heavily on the cardiovascular and immune systems, and the links between diseases of these systems and stress have been well established. Stress also has an impact of psychological well-being, with depression and post-traumatic stress disorder being the two psychiatric conditions which stress seems to exacerbate the worst. Thankfully, research has shown that the impact of stress on health can be moderated, in some cases, reversed, by learning coping strategies to effectively deal with stressors. {{Hide in print| ==Quiz== <quiz> {According to Cognitive Appraisal Theory, a stressor can be perceived as either: |type="()"} + A threat or a challenge. - A challenge or a hazard. - A hazard or a risk. - None of the above. {Approximately what percentage of adults that have suffered a traumatic life event go on to develop post-traumatic stress disorder? |type="()"} -20% +10% - greater than 50% -75% {Can emotions such as depression and anxiety affect the function of immune system cells? |type="()"} +Yes, emotional states such as depression and anxiety can have an impact on immune system function. -No, only hormones can affect the cells of the immune system. {Which of the psychological and physiological effects below is elicited during the stress reaction? (Clue: more than one answer is correct) |type="[]"} +Emotions such as anger, fear, depression and anxiety -Hunger +Increased heart rate and blood pressure -The desire to sleep {Is stress detrimental to health? |type="()"} +Yes, both chronic and acute stress have a range of impacts on health. -No, stress plays no role in health. -Yes, but only long-term exposure to chronic stress affects health. -Yes, but only work-related stress is damaging to health. </quiz> }} ==References== <div style="padding-left: 2em; text-indent: -2em"> Australian Bureau of Statistics. (2008). Causes of Death (cat.no. 3303.0). Retrieved from http://www.abs.gov.au/ausstats/abs@.nsf/Latestproducts/3303.0Main%20Features12008?opendocument&tabname=Summary&prodno=3303.0&issue=2008&num=&view= Baum, A., & Posluszny, D. M. (1999). Health psychology: Mapping biobehavioral contributions to health and illness. Annual Review of Psychology, 50, 137-163. doi:10.1146/annurev.psych.50.1.137 Bisson, J. I. (2007). 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Depression and heart disease: What do we know and where are we headed? Cleveland Clinical Journal of Medicine, 76(1), 59-70. doi: 10.3949/ccjm.75a.08011 Rubin, J., & Wessely, S. (2006). The role of stress in the etiology of medically unexplained syndromes. In B. B. Arnetz, & R. Ekman (Eds.), Stress in Health and Disease (pp. 292-305). Weinheim: Wiley-VCH. Schneider, R. H., Alexander, C. N., Salerno, J., Rainforth, M., & Nidich, S. (2005). Stress Reduction in the Prevention and Treatment of Cardiovascular Disease in African Americans: A Review of Controlled Research on the Transcendental Meditation Program. Journal of Social Behavior & Personality, 17(1), 159-180. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=17123964&site=ehost-live Schnenck-Gustatsson, K. (2009). Risk factors for cardiovascular disease in women. Maturitas, 63(3), 186-190. doi: 10.1016/j.maturitas.2009.02.014 Selye, H. (1950). Stress and the general adaptation syndrome. The British Medical Journal, 1(4667), 1383-1392. Siegrist, J. (2008). Chronic psychosocial stress at work and risk of depression: evidence from prospective studies. European Archives of Psychiatry & Clinical Neuroscience, 258, 115-119. doi:10.1007/s00406-008-5024-0 Söderpalm, B., & Söderpalm, A. (2006). The role of stress in the etiology of medically unexplained syndromes. In B. B. Arnetz, & R. Ekman (Eds.), Stress in Health and Disease (pp. 384-400). Weinheim: Wiley-VCH. Stein, M. B. (2002). Taking aim at posttraumatic stress disorder: Understanding its nature and shooting down myths. The Canadian Journal of Psychiatry, 47(10), 921-922. Sun, Y., Tao, F., Hao, J., & Wan, Y. (2010). The mediating effects of stress and coping on depression among adolescents in China. Journal of Child and Adolescent Psychiatric Nursing, 23(3), 173-180. doi: 10.1111/j.1744-6171.2010.00238.x Torii, R., Oshima, M., Kobayashi, T., Takagi, K., & Tezduyar, T. E. (2007). Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm - dependence of the effect on the aneurysm shape. Int. J. Numer. Meth. Fluids, 54, 995-1009. doi: 10.1002/fld.1497 Ueyama, T., Kasamatsu, K., Hano, T., Tsuruo, Y., & Ishikura, F. (2008). Catecholamines and estrogen are involved in the pathogenesis of emotional stress-induced acute heart attack. Stress, Neurotransmitters and Hormones: Ann. N.Y. Acad. Sci, 1148, 479-485. doi: 10.1196/annals.1410.079 Ursin, H., & Eriksen, H. (2007). Cognitive activation theory of stress, sensitization and common health problems. Ann. N.Y. Acad. Sci., 1113, 304-310. doi: 10.1196/annals.1391.024 Wang, J., Keown, L., Patten, S. B., Williams, J. A., Currie, S. R., Beck, C. A., Maxwell, C. J., & El-Guebaly, N. (2009). A population-based study on ways of dealing with daily stress: comparisons among individuals with mental disorders, with long-term general medical conditions and healthy people. Social Psychiatry and Psychiatric Epidemiology, 44(8), 666-674. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cmedm&AN=19039509&site=ehost-live Wicher, M., Myin-Germeys, I., Jacobs, N., Peeters, F., Kenis, G., Derom, C., Vlientinck, P. D., & Vas Os, J. (2007). Genetic risk of depression and stress-induced negative affect in daily life. British Journal of Psychiatry, 191, 218-223. doi: 10.1192/bjp.bp.106.032201 Yu, L., Chiu, C., Lin, Y., Wang, H., & Chen, J. (2007). Testing a model of stress and health using meta-analytic path analysis. The Journal of Nursing Research: JNR, 15(3), 202-214. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cmedm&AN=17806037&site=ehost-live Yule, W. (1999). Post-traumatic stress disorder. Arch Dis Child, 80(2), 107-109. </div> [[Category:Motivation and emotion/Book/Stress]] [[Category:Motivation and emotion/Book/Health]] h60kb2ozq9zlsk6vf7gelij6ojg616a 2816055 2816050 2026-06-17T00:39:15Z Jtneill 10242 2816055 wikitext text/x-wiki {{title|Stress and health}} {{MECR|http://screenr.com/wsc}} __TOC__ ==What is stress?== [[Image:Worried People 2.jpg|thumb|300px|right|Stress is experienced by almost all people and can be both beneficial and detrimental.]] Imagine yourself in the following situation: It is the morning of an important exam, and you are feeling worried because you think that you aren’t quite prepared. You get into your car, but have trouble starting the engine. You crank the motor again and again without any luck. Finally, the motor roars into life and you are on your way to university. You are already running a little late, but think that you can still make it to your exam on time if you drive a little faster than you probably should – until you turn onto the main road to the university to find that traffic is at a standstill. Now, think of the different physical sensations and emotions you would be feeling in the aforementioned situation. You would probably feeling anxious and worried, your heart would be racing, you would be sweating, your stomach would be turning and you would suddenly feel highly alert. These are all symptoms of the stress response. Life is inherently stressful and almost every person experiences stress at some stage in their life. [[w:Stress (biology)|Stress]] is a non-specific emotional and physiological response to demands in the environment, both positive and negative (Baum & Posluszny, 1999). In an evolutionary context, stress is an extremely important defensive reaction, activating the ‘fight or flight’ response – a process by which the body prepares to either face or run away from danger. The stress response is a multifaceted process that occurs when individuals are faced with stressors (stimuli that elicit stress) and involves both psychological and physiological elements. ==The stress response== The stress response is an automatic physical and emotional response to stimuli that is perceived to be a threat to safety. Unconsciously, the cerebral cortex alerts the [[w:hypothalamus|hypothalamus]] and the pituitary gland,which send signals to the adrenal cortex via the peripheral nervous system to release epinephrine and norepinephrine into the blood stream. Simultaneously, the pituitary gland sends signals to the adrenal cortex via adrenocorticotropic hormone (ACTH) that is released into the blood. ACTH stimulates the adrenal glands to secrete mineralocorticoids and glucocorticoids (Marieb & Hoehn, 2007; Myers, 2007). These hormones affect a number of organs and systems in the body to produce the following symptoms: * Increased heart rate * Increased blood pressure * Increase blood-sugar concentration * Widening of the arteries * Diversion of blood away from stomach * Increased blood flow to skeletal muscles * Increased muscle tension and strength (Myers; Marieb & Hoehn) The stress response also induces an emotional and cognitive reaction from the individual. Many emotions are elicited during the stress response such as anger, fear, anxiety or depressed mood (Baum & Posluszny, 1999). Cognitive function is also enhanced during the stress response, as mental alertness is heightened (Myers, 2007). All of these elements prepare the organism to either confront or run from the perceived danger. ==Theories of Stress== ===General Adaptation Syndrome (Selye)=== [[Image:General Adaptation Syndrome.jpg|thumb|400px|right|''Figure 1''. Diagram of the three stages of the General Adapation Syndrome (GAS)]] In 1950, Hans Selye published his theory explaining the mechanisms of stress – General Adaptation Syndrome (GAS) - in the British Medical Journal. This theory is still highly influential today in medical research. According to GAS, the response that every organism shows to stress is the same, regardless of the organism or the stressor that induces the reaction (Selye, 1950). In the GAS model, there are three stages to the stress response: the alarm reaction, resistance and exhaustion (see Figure 1). # The Alarm Reaction – In this stage, the stressor is identified by the organism and the stress response rapidly occurs. Initially, the ability to resist the stressor is diminished, due to the physiological effects caused by the activation of the sympathetic nervous system (such as increased heart rate and diversion of blood away from organs such as the stomach to the skeletal muscles) (Myers, 2007). However, the body quickly adjusts and moves into stage two, resistance (Selye, 1950). # Resistance – During this stage, the body deals with the stressor by increasing heart rate and blood pressure, increasing energy available to the skeletal muscles and heightening alertness temporarily (Marieb & Hoehn, 2007). This response is only designed to be sustained for a short period of time, after which stage 3 occurs (Selye, 1950). # Exhaustion – In stage 3 of GAS, if the presence of the stressor is sustained, the body’s resources become depleted and the body can no longer sustain an appropriate level of resistance to the stressor. The ability of the body to deal with the stressor becomes lower than it was prior to the stress response, which can leave the organism vulnerable to injury or disease (Myers, 2007; Selye, 1950). Whilst this model is widely accepted by the medical community as an appropriate conceptualisation of the stress response, Selye did not make any firm distinction between the different type of stressors and did not take into account individual differences in the way people perceive stressors (Lazarus, 1993). In this model, both psychological and physiological stressors are treated in the same way; that is, a stressor is defined as anything that is potentially damaging to tissue (Lazarus, 1993). However, in reality, individual perceptions on what is a threat and what is harmful vary largely between different people (McCraty & Tomasino, 2006). In reaction to this, research moved in the direction of explaining the psychological (particularly, the emotional and cognitive) component of the stress response. ===Cognitive Appraisal Theory (Lazarus)=== [[Image:Studying.jpg|260px|right|thumb|Studying for an exam is a stressor that is often viewed as a challenge, rather than a threat.]] The cognitive appraisal theory looks to address some of the deficits of the GAS theory and centres around the concept that it is the organism's perception and appraisal of the stressor that is the most important factor in the intensity of the stress response. This theory suggests that the stress response can be mediated by the individuals appraisal of the stressor as either a ''threat'' or a ''challenge'' (Larazus, 1990). Depending on whether the stressor is considered a threat to safety or positive challenge, the body’s reaction will differ. If the stressor is appraised to be a challenge, the stress response will be diminished and the experience will be more positive than if the stressor is perceived as a threat (which would induce the ‘typical’ stress response, defined by GAS) (Lazarus, 1993). Lazarus (1990) argues that the stress response is just one part of a larger process that is driven by cognition. This theory has been received favourably given that it takes into account the large psychological component of stress and how individual experience plays a role in the perception of stressors (Ursin & Eriksen, 2001). This theory also have implications for the management of stress, which is discussed later in this chapter. ==Relationship between Stress and Health== [[Image:Stress 2.gif|thumb|400px|left|''Figure 2''. Chronic stress has a variety of effects on the body]] Stress is one of the most important factors that mediates the relationship between [[w:Health|health]] and [[w:Behavior|behaviour]] (Baum & Posluszny, 1999). It is widely regarded as one of the most important factors in illness (Hanson & Chen, 2010) and is fast becoming a major public health problem (Gehling, Aubert, Padlina, Martin-Diener & Somaini, 2001). Acute stress can be beneficial for an organism, triggering the ‘fight or flight’ response and thereby increasing strength, stamina and attention for a short period of time (Ekman & Arnetz, 2006). However, chronic stress is disruptive to a multitude of psychological and physiological processes. Over time, if the stress response system is activated continuously, the allostatic load caused by this can put strain on the body (Hanson & Chen, 2010). The ‘exhaustion’ stage of Selye’s GAS theory states that prolonging and sustaining the physiological and psychological reactions of stress long-term has the potential to weaken the body’s immune responses through exhaustion, thereby predisposing the body to disease (Baum & Posluszny, 1999; Selye, 1950). In the long-term, the constant presence of stress hormones within the body results in a range of effects on the organs that create homeostatic imbalance. Mineralocorticoids cause the retention of water and sodium in the body by the kidneys, which increases blood volume and blood pressure. Glucocorticoids increase the amount of glucose in the blood by suppressing insulin and increasing the conversion of fat to glucose (Marieb & Hoehn, 2007). Both of these processes put strain on the cardiovascular and immune systems, as both of these systems use the blood to transport oxygen, antibodies, immune cells and nutrients to cells as well as transporting waste and infection away from cells. The psychological impacts of stress also play a role in health outcomes. Research has shown that individuals who are under chronic stress are more likely to engage in unhealthy behaviours such as excessive drinking, drug misuse, smoking, overeating and sedentary behaviour in order to manage and relieve the psychological discomfort of stress (Krueger & Chang, 2008). Increased alcohol and drug misuse has been strongly linked to stress in the research literature. The affect regulation model suggests that when stress increases, individuals 'self-medicate' with alcohol and drugs in order to reduce the feelings of distress and discomfort that come with stress (Grzywacz & Almeida, 2008). In their study into alcohol use, negative mood and stress, Grzywacz and Almeida found that the probability that a person would engage in binge drinking behaviours was largely increased on days when individuals experienced stress. Smokers also tend to report increasing their number of cigarettes they smoke during periods of high stress (Söderpalm & Söderpalm, 2006). Engaging in unhealthy behaviours long-term can result in a number of poor health outcomes such as cancer, obesity, health complications related to drug dependence or misuse and death (Krueger & Chang, 2008). Chronic stress also leads to more instances of self-reported poor health and has been shown to be a large contributor to psychosomatic illness or pain (Ursin & Eriksen, 2007). Numerous studies have found that individuals who suffer from chronic stress report significantly poorer health than individuals who do not suffer chronic stress (Holmgren, Dahlin-Ivanoff, Björnlund & Hensing, 2009; Yu, Chiu, Lin, Wang & Chen, 2007). A study of Swiss participants found that not only did those who reported suffering chronic stress report more health complaints, the severity of the symptoms that they reported was significantly more likely to fall within a clinical range (Gehling, Aubert, Padlina, Martin-Diener & Somaini, 2001). Stress also appears to play a large role in the onset of psychosomatic illness, pain and medically unexplained health conditions. Sufferers of medically unexplained syndromes (such as chronic fatigue syndrome) often report that a stressful event preceded the onset of their condition (Rubin & Wessely, 2006). ==Physiological Impacts of Stress== ===Cardiovascular=== According to figures released by the Australian Bureau of Statistics, cardiovascular disease was the number one cause of death for both males and females in Australia during 2008 (ABS, 2010). [[w:Cardiovascular disease|Cardiovascular disease]] (CVD) is a blanket term used to describe a range of disorders that affect different components of the cardiovascular system including conditions such as myocardial infarction (heart attack), stroke, hypertension (high blood pressure), arrhythmia, aneurysm, thrombosis and oedema (Marieb & Hoehn, 2007). The associations between emotional states (such as stress) and cardiovascular disease have been well established over the last 20 years and have been shown to influence the development of cardiovascular pathology independent of other risk factors like smoking, weight and diet (Grippo & Johnson, 2009). So unsurprisingly, understanding the effects of stress on cardiovascular disease is becoming an important priority for medical science. [[Image:Heart.jpg|thumb|200px|left|The heart can be damaged by chronic stress.]] The relationship between stress (both acute and chronic) and CVD is well established, particularly in relation to myocardial infarction. Schnenck-Gustatsson (2009) found that stress was one of the most important risk factors in the development of CVD in women. In fact, stress is the primary cause of one type of myocardial infarction, called takotsubo cardiomyopathy, which only affects post-menopausal women (Ueyama, Kasamatsu, Hano, Tsuruo & Ishikura, 2008). This particular type of myocardial infarction was identified after an earthquake occurred in Japan, and the number of women suffering heart attacks increases dramatically (Ueyama, Kasamatsu, Hano, Tsuruo & Ishikura, 2008). As well as major life events, such natural disasters or periods of change and uncertainty, personality can influence stress, and it has been investigated in relation to CVD. During the 1980’s, type A personality was linked with CVD. Individuals with type A personalities tend to experience more stress due their propensity to be hostile and aggressive (Friedman & Booth-Kewley, 1987). However, research into type A personality and CVD was mixed and inconclusive, and the focus in recent years has shifted to other personality types that may be related to stress, such as type D 'depressive' personality (Pozuelo, Zhang, Franco, Tesar, Penn & Jiang, 2009; Denollot, 1998). Given that a large portion of the changes that occur during the stress response are targeted at the components of the cardiovascular system such as the veins and arteries, it is unsurprising that over time, damage can also occur to these components of the cardiovascular system. High blood pressure (which is one of the primary symptoms of stress) when experienced over the long term, can weaken the walls of blood vessels and cause aneurysms to develop and rupture (Torii, Oshima, Kobayashi, Takagi & Tezduyar, 2007). Links have also been found between the presence of severe, acute stressors and the development of heart arrhythmias (Grippo & Johnson, 2009). One theme that appears from the research literature into CVD and stress is that stress is an independent risk factor (given the pressure that the stress response directly puts on the cardiovascular system), but it also exacerbates other risk factors such as diet, smoking, levels of physical activity and weight. As mentioned previously, stress impacts the likelihood of a person engaging in unhealthy behaviours (Krueger & Chang, 2008). Learning to manage stress is one of the treatment strategies taught to CVD patients to help them manage their condition and reduce the likelihood of future illness (Grippo & Johnson, 2009). {| cellpadding="10" cellspacing="5" style="float: right; width: 40%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 40%; background-color:LightGreen; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 10px;" | {{center top}} {{big2|'''Case Study: Part One'''}} {{center bottom}} [[Image:Migraine.jpg|100px|left]] Jayne is a young woman in her early 20's who suffers from chronic stress. She reports feeling stressed about several aspects of her life, including her work-life balance, her university studies as well as conflicts that have occurred between herself and some friends. Over the past 12 months, she has also noticed that her health has been suffering. Jayne has had a number of illnesses, including some recurring problems. She has had four bouts of gastroenteritis, two ear infections, several bouts of common cold as well as one bout of 'flu and a severe case of whooping cough (even though she had received vaccinations for both). She reports being 'constantly sick' and 'lacking energy'. Jayne feels as though her poor health is exacerbating her stress and is desperate to get her health back under control. |} ===Infectious Diseases=== The situation experienced by the young woman in the case study, Jayne, is not uncommon for people who suffer chronic stress. Stress has a clear impact on the [[w:Immune system|immune system]] and the wear-and-tear caused by stress can leave the body unable to mount a defense against incoming pathogens (Godbout & Glaser, 2006; Baum & Posluszny, 1999). Infectious diseases such as upper-respiratory infections, the 'flu, strains of herpes virus and hepatitis C have all been shown to spread faster in individuals experiencing chronic stress (Cohen, Janicki-Deverts & Miller, 2007). The presence of chronic stress can increase the duration and severity of infectious diseases (Cohen, Janicki-Deverts & Miller, 2007). It has also been shown to help provide an environment within the body that promotes the activation and duplication of viruses and bacteria and is also being investigated as a potential cofactor in tumour growth and development (Godbout & Glaser, 2006). One of the hormones that is released during the stress response, glucocorticoid, has been implicated as the potential primary cause of immuno-suppression (Godbout & Glaser,). Once the immune system has been suppressed, viruses and bacteria that cause infectious diseases are more easily able to multiple and create infection within the body (Baum & Posluszny, 1999). The emotional state that stress induces is also implicated in the increased propensity for those suffering chronic stress to also develop infectious diseases more rapidly (Clougherty & Kubzansky, 2009). The emotions associated with stress, such as depression and anxiety, has been shown to reduce the activity of key immune system cells such as natural killer cells and T- and B-cells (Godbout & Glaser, 2006). The suppressed state of the immune system that chronic stress induces has large implications for the management and prevention of infectious diseases. Wound healing and surgical recovery has been shown to be greatly impacted if patients are in stressful environments. Godbout and Glaser (2006) found that wound healing was far slower under stressful conditions and the potential for the wound becoming infected was largely increased. The prevention of infectious diseases can be impaired by presence of chronic stress. When reading the case study, you may have asked yourself 'how it is that Jayne could have caught the flu and whooping cough, given that she had received vaccinations against those two viruses?'. Research shows that if Jayne received these inoculations whilst she was suffering chronic stress, they may not have worked. The antibody development process that occurs after being vaccinated can be impaired if the vaccine is administered whilst the recipient is under stress (Godbout & Glaser, 2006). B-cells play an essential role in the development of antibodies, and when the body is under psychological stress, the function of these cells is greatly reduced (Clougherty & Kubzansky, 2009). ==Psychological Impacts of Stress== ===Depression=== The stress response elicits a wide range of emotions, however, it is mainly responsible for producing a state of negative affect (Krueger & Chang, 2008). Stress is considered to be one of the primary risk factors for the ontset of [[w:Major depressive disorder|major depressive disorder]] and episodes of sub-clinical depression (Husky, Mazure, Maciejewski & Swendsen, 2008). Husky et al. found that stress has a large influence on the development of an initial episode of depression, however tends to play a lesser role in every subsequent episode of depression. This finding is consistent with much of the research literature on the role of stress in depression. One study found that in the three to six months preceding the onset of depressive symptoms, 50 to 80% of depressed individuals reported experiencing at least one stressful life event (Cohen, Janicki-Deverts & Miller, 2007). The genetic predisposition to develop depression also appears to play a role in how individuals react to stressors. A recent study found that those with a family history of depression tended to report heightened negative affect in reaction to life stressors (Wichers et al., 2007). There appears to be an interplay between genetics, cognitive and environmental factors in the relationship between stress and depression. No one type of stressor appears to have a greater influence on the development of depression than any other. In a study into the effects of work-place stress on mental health, Siegrist (2008) found that work-related stress was associated with a significantly increased risk of developing depression. Childbirth is also a highly stressful life event, and post-natal depression can develop during this period of high stress and rapid change. Interestingly, the stress of having a new baby appears to influence the development of depressive symptoms equally in both mothers and fathers (Gao, Chan & Mao, 2009). However, Gao et al. (2009) found that mothers tended to be able to cope with the stress better, due to greater social support. The ability to cope with the stress appeared to mediate the level of depression that the mothers in Gao’s study suffered. Research, like the Gao et al. study, suggest that one of the factors that may contribute to the high prevalence of depressive disorders amongst adolescents and young adults may be their inability to cope with stress effectively. Sun, Tao, Hao and Wan (2010) found in a study of Chinese adolescents that the ability to cope with stressful life events reduced the likelihood of developing depression. ===Post-traumatic Stress Disorder=== [[Image:Post-traumatic stress disorder world map - DALY - WHO2004.svg|right|thumb|300px|This map shows the countries with the highest prevalence of PTSD in red.]] Post-traumatic stress disorder (PTSD) is arguably the one of the worst health outcomes that comes from exposure to stress. PTSD is a debilitating psychiatric condition that is triggered by a period of highly traumatic and stressful experiences that involved actual or threatened serious injury or death or the actual or threatened danger to personal integrity, either of the individual or others (Bisson, 2007). PTSD is most prevalent among males, and there is a high prevalence amongst veterans of war (Bisson, 2007; Stein, 2002). Following the stressful event, sufferers of PTSD experience severe and debilitating stress reactions whether or not they are in the presence of a stimuli that may remind them of the traumatic event (Yule, 1999). The stress reactions that sufferers experience when they experience flashbacks are linked with the development of depression and suicide attempts are common amongst those with PTSD who are of adolescent age and over (Yule, 1999). The neurological damage that is associated with long-term chronic stress (such as decreased hippocampus size, due to neuron atrophy) can be found in sufferers of PTSD (Stein, 2002). Even though greater than 50% of adults have experienced at least one traumatic event during their life time, only approximately 10% of these individuals suffer from PTSD (Bisson, 2007). Exactly what makes one individual more susceptible to PTSD than another is still an area of debate in psychology. One model suggests that suffering a psychological disorder earlier in life increases the chance of developing PTSD after being exposed to a traumatising stressor (Cohen, Matar, Richter-Levin & Zohar, 2006). Debate about what biological mechanisms drive PTSD is also still up for debate, although dysfunction of the same hypothalamic-pituitary-adrenal axis that is activated in the normal stress response is returning promising results in an animal model (Cohen et al., 2006). ==Preventative measures== {| cellpadding="10" cellspacing="5" style="float: left; width: 40%; background-color: inherit; margin-left: auto; margin-right: auto" | style="width: 40%; background-color:LightGreen; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px; height: 10px;" | {{center top}} {{big2|'''Case Study: Part Two'''}} {{center bottom}} [[Image:Migraine.jpg|100px|left]] Jayne decided to seek help from a psychologist in order to get her stress levels under control. She was taught strategies to help her manage her stress and began exercising and mediating in order to keep stress levels under control. Jayne also stopped engaging in unhealthy behaviours, like sedentary behaviour, when she began to feel stressed. As her stress level declined over the next six months, her health improved. She was sick less frequently and began to feel more energetic. |} Given the wide variety of health problems that are associated with stress, understanding how to manage stress in order to improve health is a high priority, particularly in developed nations (Gehring, Aubert, Padlina, Martin-Diener & Somaini, 2001). Thankfully, research is beginning to reveal that the effect of stress on health can be mediated by an individual's ability to cope with the stressor (Wang, Keown, Patten, Williams, Currie, Beck, Maxwell and El-Guebaly, 2009). According to the cognitive appraisal theory, it is the perception of the stressor that has the greatest impact on the level of stress produced (Lazarus, 1990), so learning strategies to help change the way an individual perceives everyday stressors (like the traffic jam from the start of the chapter) has a large impact on the levels of stress an individual experiences (Yu, Chiu, Lin, Wang & Chen, 2007). Like Jayne, many individuals use physical exercise as an effective method of reducing stress, although it can be difficult to get people who are stressed to engage in physical activity as sedentary behaviour can be highly rewarding over the short-term (Siegrist, 2008; Baum & Posluszny, 1999). Mediation has also been shown to be another effective method of stress reduction. In their study into CVD in an African-American population, Schneider and his colleagues found the use of transcendental mediation not only reduced stress, but in doing so, it significantly reduced the risk of CVD (Schneider, Alexander, Salerno, Rainforth & Nidich, 2005). Reducing stress can help improve self-reported health, as in the case of Jayne. Given the fact that the body is not designed to deal with chronic stress (Grippo & Johnson, 2009), making sure stress levels are managed is fast become an accepted method of controlling one of today's the largest public health issues. ==Summary== Stress is an often unwelcome but common part of our lives and is widely recognised as one of the primary risk factors in health and illness (Hanson & Chen, 2010). This chapter has discussed the relationship between stress and health, looked in depth at four health conditions in which stress is a primary risk factor and discussed some of the preventative methods that can be used to reduce stress and thereby improve health. The two most prominent theories of stress - GAS and the Cognitive Appraisal theory - focus on different aspects of the stress response (the physiological and the psychological), but have both been shown to be quite useful in understanding the multifaceted nature of stress. The physiological features of the stress response impact heavily on the cardiovascular and immune systems, and the links between diseases of these systems and stress have been well established. Stress also has an impact of psychological well-being, with depression and post-traumatic stress disorder being the two psychiatric conditions which stress seems to exacerbate the worst. Thankfully, research has shown that the impact of stress on health can be moderated, in some cases, reversed, by learning coping strategies to effectively deal with stressors. {{Hide in print| ==Quiz== <quiz> {According to Cognitive Appraisal Theory, a stressor can be perceived as either: |type="()"} + A threat or a challenge. - A challenge or a hazard. - A hazard or a risk. - None of the above. {Approximately what percentage of adults that have suffered a traumatic life event go on to develop post-traumatic stress disorder? |type="()"} -20% +10% - greater than 50% -75% {Can emotions such as depression and anxiety affect the function of immune system cells? |type="()"} +Yes, emotional states such as depression and anxiety can have an impact on immune system function. -No, only hormones can affect the cells of the immune system. {Which of the psychological and physiological effects below is elicited during the stress reaction? (Clue: more than one answer is correct) |type="[]"} +Emotions such as anger, fear, depression and anxiety -Hunger +Increased heart rate and blood pressure -The desire to sleep {Is stress detrimental to health? |type="()"} +Yes, both chronic and acute stress have a range of impacts on health. -No, stress plays no role in health. -Yes, but only long-term exposure to chronic stress affects health. -Yes, but only work-related stress is damaging to health. </quiz> }} ==References== <div style="padding-left: 2em; text-indent: -2em"> Australian Bureau of Statistics. (2008). Causes of Death (cat.no. 3303.0). Retrieved from http://www.abs.gov.au/ausstats/abs@.nsf/Latestproducts/3303.0Main%20Features12008?opendocument&tabname=Summary&prodno=3303.0&issue=2008&num=&view= Baum, A., & Posluszny, D. M. (1999). Health psychology: Mapping biobehavioral contributions to health and illness. Annual Review of Psychology, 50, 137-163. doi:10.1146/annurev.psych.50.1.137 Bisson, J. I. (2007). Post-traumatic stress disorder. Occupational Medicine, 57, 399-403. doi:doi:10.1093/occmed/kqm069 Clougherty, J. E., & Kubzansky, L. D. (2009). A Framework for Examining Social Stress and Susceptibility to Air Pollution in Respiratory Health. Environmental Health Perspectives, 117(9), 1351-1358. doi:10.1289/ehp.0900612 Cohen, S., Janicki-Deverts, D., & Miller, G. E. (2007). Psychological stress and disease. JAMA, 298(14), 1685-1687. doi:10.1001/jama.298.14.1685 Denollet, J. (1998). Personality and coronary heart disease: The type-D scale 16 (DS16). Ann Beh Med, 20(3), 209-215. Ekman, R., & Arnetz, B.B. (2006). The brain in stress – Influence of environment and lifestyle on stress-related disease. In B. B. Arnetz, & R. Ekman (Eds.), Stress in Health and Disease (pp. 280-290). Weinheim: Wiley-VCH. Friedman, H. S., & Booth-Kewley, S. (1987). Personality, type A behaviour and coronary heart disease: The role of emotional expression. Journal of Personality and Social Psychology, 53(4), 783-792. Gao, L., Chan, S. W., & Mao, Q. (2009). Depression, perceived stress and social support among first-time Chinese mothers and fathers in the postpartum period. Research in Nursing and Health, 32, 50-58. Gehring, T. M., Aubert, L., Padlina, O., Martin-Diener, E., & Somaini, B. (2001). Perceived stress and health-related outcomes in a Swiss population sample. Swiss Journal of Psychology/Schweizerische Zeitschrift Für Psychologie/Revue Suisse De Psychologie, 60(1), 27-34. doi:10.1024//1421-0185.60.1.27 Godbout, J. P., & Glaser, R. (2006). Stress-induced immune dysregulation: Implications for wound healing, infectious disease and cancer. J Neuroimmune Pharm, 1, 421-427. doi: 10.1007/s11481-006-9036-0 Grippo, A. J., & Johnson, A. K. (2009). Stress, depression and cardiovascular dysregulation: a review of neurobiological mechanisms and the integration of research from preclinical disease models. Stress (Amsterdam, Netherlands), 12(1), 1-21. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cmedm&AN=19116888&site=ehost-live Grzywacz, J. G., & Almeida, D. M. (2008). Stress and binge drinking: A daily process examination of stressor pile-up and socioeconomic status in affect regulation. International Journal of Stress Management, 15(4), 364-380. doi:10.1037/a0013368 Hanson, M. D., & Chen, E. (2010). Daily stress, cortisol, and sleep: The moderating role of childhood psychosocial environments. Health Psychology, 29(4), 394-402. doi:10.1037/a0019879 Holmgren, K., Dahlin-Ivanoff, S., Björkelund, C., & Hensing, G. (2009). The prevalence of work-related stress and its association with self-perceived health and sick-leave, in a population of Swedish women. BMC Public Health, 9(73) doi:10.1186/1471-2458-9-73 Husky, M. M., Mazure, C. M., Maciejewski, P. K., & Swendsen, J. D. (2009). Past depression and gender interact to influence emotional reactivity to daily life stress. Cognitive Therapy and Research, 33(3), 264-271. doi:10.1007/s10608-008-9212-z Krueger, P. M., & Chang, V. W. (2008). Being poor and coping with stress: Health behaviours and the risk of death. Am J Public Health, 98, 889-896. doi:doi:10.2105/AJPH.2007.114454 Lazarus, R. S. (1990). Theory-based stress management. Psychological Inquiry, 1(1), 3-13. Lazarus, R. S. (1993). From psychological stress to the emotions: A history of changing outlooks. Annu. Rev. Psychol., 44, 1-21. McCraty, R., & Tomasino, D. (2006). Emotional stress, positive emotions and psychophysiological coherence. In B. B. Arnetz, & R. Ekman (Eds.), Stress in Health and Disease (pp. 342-364). Weinheim: Wiley-VCH. Myers, D. G. (2007). Psychology (8th ed.). New York, NY: Worth Publishers. Pozuelo, L., Zhang, J., Franco, K., Tesar, G., Penn, M., & Jiang, W. (2009). Depression and heart disease: What do we know and where are we headed? Cleveland Clinical Journal of Medicine, 76(1), 59-70. doi: 10.3949/ccjm.75a.08011 Rubin, J., & Wessely, S. (2006). The role of stress in the etiology of medically unexplained syndromes. In B. B. Arnetz, & R. Ekman (Eds.), Stress in Health and Disease (pp. 292-305). Weinheim: Wiley-VCH. Schneider, R. H., Alexander, C. N., Salerno, J., Rainforth, M., & Nidich, S. (2005). Stress Reduction in the Prevention and Treatment of Cardiovascular Disease in African Americans: A Review of Controlled Research on the Transcendental Meditation Program. Journal of Social Behavior & Personality, 17(1), 159-180. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=17123964&site=ehost-live Schnenck-Gustatsson, K. (2009). Risk factors for cardiovascular disease in women. Maturitas, 63(3), 186-190. doi: 10.1016/j.maturitas.2009.02.014 Selye, H. (1950). Stress and the general adaptation syndrome. The British Medical Journal, 1(4667), 1383-1392. Siegrist, J. (2008). Chronic psychosocial stress at work and risk of depression: evidence from prospective studies. European Archives of Psychiatry & Clinical Neuroscience, 258, 115-119. doi:10.1007/s00406-008-5024-0 Söderpalm, B., & Söderpalm, A. (2006). The role of stress in the etiology of medically unexplained syndromes. In B. B. Arnetz, & R. Ekman (Eds.), Stress in Health and Disease (pp. 384-400). Weinheim: Wiley-VCH. Stein, M. B. (2002). Taking aim at posttraumatic stress disorder: Understanding its nature and shooting down myths. The Canadian Journal of Psychiatry, 47(10), 921-922. Sun, Y., Tao, F., Hao, J., & Wan, Y. (2010). The mediating effects of stress and coping on depression among adolescents in China. Journal of Child and Adolescent Psychiatric Nursing, 23(3), 173-180. doi: 10.1111/j.1744-6171.2010.00238.x Torii, R., Oshima, M., Kobayashi, T., Takagi, K., & Tezduyar, T. E. (2007). Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm - dependence of the effect on the aneurysm shape. Int. J. Numer. Meth. Fluids, 54, 995-1009. doi: 10.1002/fld.1497 Ueyama, T., Kasamatsu, K., Hano, T., Tsuruo, Y., & Ishikura, F. (2008). Catecholamines and estrogen are involved in the pathogenesis of emotional stress-induced acute heart attack. Stress, Neurotransmitters and Hormones: Ann. N.Y. Acad. Sci, 1148, 479-485. doi: 10.1196/annals.1410.079 Ursin, H., & Eriksen, H. (2007). Cognitive activation theory of stress, sensitization and common health problems. Ann. N.Y. Acad. Sci., 1113, 304-310. doi: 10.1196/annals.1391.024 Wang, J., Keown, L., Patten, S. B., Williams, J. A., Currie, S. R., Beck, C. A., Maxwell, C. J., & El-Guebaly, N. (2009). A population-based study on ways of dealing with daily stress: comparisons among individuals with mental disorders, with long-term general medical conditions and healthy people. Social Psychiatry and Psychiatric Epidemiology, 44(8), 666-674. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cmedm&AN=19039509&site=ehost-live Wicher, M., Myin-Germeys, I., Jacobs, N., Peeters, F., Kenis, G., Derom, C., Vlientinck, P. D., & Vas Os, J. (2007). Genetic risk of depression and stress-induced negative affect in daily life. British Journal of Psychiatry, 191, 218-223. doi: 10.1192/bjp.bp.106.032201 Yu, L., Chiu, C., Lin, Y., Wang, H., & Chen, J. (2007). Testing a model of stress and health using meta-analytic path analysis. The Journal of Nursing Research: JNR, 15(3), 202-214. Retrieved from http://ezproxy.canberra.edu.au/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cmedm&AN=17806037&site=ehost-live Yule, W. (1999). Post-traumatic stress disorder. Arch Dis Child, 80(2), 107-109. </div> [[Category:Motivation and emotion/Book/Stress]] [[Category:Motivation and emotion/Book/Health]] 5pjnwymygnh5j01gwjb3q6q40dcxjpe 10 122195 2816072 2720561 2026-06-17T05:45:51Z Jtneill 10242 Update for 2026 2816072 wikitext text/x-wiki #REDIRECT [[Template:MEBF/2026]] muewmn32zojc2fbtujk9i32membnnsu 0 139384 2815937 2815799 2026-06-16T13:41:38Z Young1lim 21186 /* Adder */ 2815937 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.2A.CLA.20260616.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260616.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]] 8zhzeu4550tt6njo4ohn3dhdazjujfh 2 145726 2815935 2813366 2026-06-16T13:31:22Z Atcovi 276019 2815935 wikitext text/x-wiki ==Atcovi/to do== === Current Projects (2026) === * [[Intuitive Calculus]] * [[User:Atcovi/OGM & Suicide/The Paper]] - ''[moved]'' * [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal. * [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]]) ====Future Endeavors==== * [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C47&as_ylo=2020&as_yhi=2025&q=Suicide+in+Sweden+refugees&btnG=] * Get [[User:Atcovi/Spring2024]] & [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] & [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]]. * Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing. =====Wikiversity-Related Works===== * Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes. **Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]? * [[Wikiversity:Original research and scholarly standards]] & improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] & [[Wikiversity:Research ethics]]. ** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? * [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. * [[Wikiversity:Artificial intelligence]] - "substantial"? What defines "substantial"? * Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. {{Archive box| {{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}} ---- {{center top}}'''Archives'''{{center bottom}} *[[User:Atcovi/to do/Current Projects/2023]] *[[User:Atcovi/to do/Current Projects/January 4, 2022]] *[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]] *[[User:Atcovi/to do/Current Projects/2015]] ---- }} [[Category:Atcovi's Work]] kb8rkjboyymfcjtmfccwr57zn4je3ii 2816049 2815935 2026-06-16T23:59:04Z Atcovi 276019 +[[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]] 2816049 wikitext text/x-wiki ==Atcovi/to do== === Current Projects (2026) === * [[Intuitive Calculus]] * [[User:Atcovi/OGM & Suicide/The Paper]] - ''[moved]'' * [[User:Atcovi/Journey to Clinical PhD]] - figuring this out; current life goal. * [[WikiJournal Preprints/Mental health in Sri Lanka]] (and later in August: [[User:Atcovi/APA2026 Abstract]]) ** [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]]. ====Future Endeavors==== * [[WikiJournal Preprints/Suicide amongst refugees in Sweden]] [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C47&as_ylo=2020&as_yhi=2025&q=Suicide+in+Sweden+refugees&btnG=] * Get [[User:Atcovi/Spring2024]] & [[User:Atcovi/Psychopathology]] into the mainspace. Develop [[Child psychology]] & [[User:Atcovi/PSYC318W]] into a complete course. Merge [[Validity]] into [[User:Atcovi/PSYC318W|PSYC318W]]. * Develop resources related to [[suicidology]] (3 stress response systems? effects of catecholamines on suicidal ideation? neurobiology of suicidal ideation? relation between autobiographical memory and suicide?), expand [[wikipedia:Suicidology#Theories_of_suicide|Suicidology#Theories_of_suicide]] either through [[WikiJournal of Science]] or WP editing. =====Wikiversity-Related Works===== * Promote [[Help:Project boxes]], something very useful and unique to Wikiversity. Focus on trying to not only create more project boxes, but to define resource types used in project boxes. **Ex, what is a [[:Category:Workshops|workshop]]? What differentiates between an [[Help:Essay|essay]] and a [[Help:Paper|paper]]? What differentiates between a [[Template:Notes|notes resource]] (that may be ''derived'' from a homework assignment) and a [[Help:Assignment|homework assignment]] [small note: this page seems to be created by accident and may need a revamp]? * [[Wikiversity:Original research and scholarly standards]] & improvements/proposals for [[Wikiversity:Original research]] (ex, [[Template:Original research]] should be a mandatory addition to original research on WV + a notice letting readers know that the work is not established science). Develop other pages related to research ethics, including [[Wikiversity:Research]] & [[Wikiversity:Research ethics]]. ** [[Wikiversity:Review board]] - should this be Wikiversity 'crats that review original research proposals? * [[Wikiversity:Verifiability]] - start heavily scrutinizing pages that don't meet this criteria. * [[Wikiversity:Artificial intelligence]] - "substantial"? What defines "substantial"? * Expand [[Wikiversity:Differences between Wikiversity and Wikipedia]]. {{Archive box| {{center top}}'''[[User:Atcovi/to do|To do list]]'''{{center bottom}} ---- {{center top}}'''Archives'''{{center bottom}} *[[User:Atcovi/to do/Current Projects/2023]] *[[User:Atcovi/to do/Current Projects/January 4, 2022]] *[[User:Atcovi/to do/Current Projects/September 2017 - January 2018]] *[[User:Atcovi/to do/Current Projects/2015]] ---- }} [[Category:Atcovi's Work]] 0ra5zs5hm3jc4oui6p6f0w08lp879to 0 151889 2816098 1697675 2026-06-17T11:35:09Z ~2026-35482-89 3094950 /* Description and History */ spelling mistakes. 2816098 wikitext text/x-wiki [[The Varanasi Heritage Dossier/Detailed description of each heritage Site|Detailed description of each heritage Site]] - [[The Varanasi Heritage Dossier/Prahalad Ghat to Raj Ghat| Prahalad Ghat to Raj Ghat]] ---- [[The Varanasi Heritage Dossier/Ganje-Shahida Mosque at Raj Ghat|Ganje, Shahida Mosque at Raj Ghat]] === Location === 25º 19.463’ North and 83º 01.622’ East (Ganje-Shahida Mosque). ===== Exact location on a map ===== The old fort area, west of the Kashi Railway station. ==== Area ==== 0.03ha (the mosque compound) === Historical/cultural/natural significance === During the period of Delhi Sultanate, the message of Islam spread by the messengers (pupils) and priests (maulvis). Such people were known for faith healing and kept them always ready to be sacrificed for the good cause to serve the humanity. They were called shahida (martyr). The City of Varanasi if full of such shahidas. Faith seekers and those who feel to have faith healing, invariably of caste, creed or religion visit these places. In such place around forty per cent visitors are from the Hindu community, of course most of them poor and low caste. The Ganje-Shahida mosque is one of such examples. As early as in later part of 19th century, it has already been proved that this mosque was developed at several stages with several alterations using columns, pillars and stone slabs of an ancient temple. === Description and History === Situated opposite Kashi Railway Station in the old fort restored in the second half of the 15th century, the Ganje-Shahida Mosque was built, probably in the 13th century. In 1857 it was in a state of abandon, but has since been refurbished and returned to its religious function. Towards 1890, probes revealed the existence of a slab floor about 30cm below the present floor-level. In fact, there is a twin pair of mosques, the one more to the south having a higher ceiling and a minbar. It is impossible to tell which is older from an examination of what is now visible. The double hall is oblong with three galleries running parallel to the façade which faces east and is longer than the sides; the structure is of pillars, lintels and slabs. Two mehrab, one for each mosque, are cut into the qibla wall and are not of sandstone with the outer face much longer than high. The ruins subsisting in the 19th century have been poorly restored. The whole building is covered with a terrace reached from stairs added to the south side-wall and there is no dome. The entrance, the lavatories and the surrounding wall are recent additions. Closeby to the Ganje-Shahida Mosque, is a Mazar (tomb) of Sheikh Shah Dainas, who came here to teach the Islamic message of brotherhood in 18th century. === Present state of conservation === There is no such specific organisation to take care of preservation and conservation. === Specific measures being taken for conserving the specific property === No specific measures are taken to conserve and preserve the area. === Ownership === The area is administered and managed by the Sunni (Muslim) Wakhf (or Waquf) Board, a charitable trust, and represented by Mutawalli, a representative, who serves as Priest-cum-Administrator. {{CourseCat}} ta9syc20qwnrimnlcl43ztr2tyhltl8 0 171005 2815946 2815804 2026-06-16T14:15:27Z Young1lim 21186 /* Geometric Series Examples */ 2815946 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.20260615.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]] cqdlcz2p9qxhpfov8w98wrp4mz807s0 2815951 2815946 2026-06-16T14:21:50Z Young1lim 21186 /* Geometric Series Examples */ 2815951 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.20260616.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]] 98w4n2b6ofybe97ie14h9gmbw1se6bj 0 203942 2815974 2815087 2026-06-16T16:46:18Z Young1lim 21186 /* Lambda Calculus */ 2815974 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260615.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] bqhn2of2sxzicbyxz59389n0df13mn2 2815976 2815974 2026-06-16T16:48:21Z Young1lim 21186 /* Lambda Calculus */ 2815976 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260616.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] qgojc9vcea08yz6wfslf51r164yvph2 2816096 2815976 2026-06-17T10:58:54Z Young1lim 21186 /* Lambda Calculus */ 2816096 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260617.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] 1ljqznq4xe06xx9vf2kp3ijevc1uv51 1 206532 2815961 2784463 2026-06-16T15:24:13Z Regliste 3029369 /* Is Wikijournal of Science still active? */ Reply 2815961 wikitext text/x-wiki {{WikiJournal_discussions|Crab Nebula (crop2).jpg}} {{Archive box| [[/2016-2017|2016-2017]] Discussions may also take place at the <br>'''[https://groups.google.com/forum/#!forum/wikijsci/join public mailing list]''' }} [[Category:WikiJournal of Science]] {{TOClimit|limit=3}} == WikiJSci year summary == [[File:WJS Poster.pdf|thumb|WikiJSci poster]] Thank you to everyone for this year at WikiJSci. The [[WikiJournal of Science/Editorial board|editorial board]] is growing to cover a range of expertise. There have been very useful discussions on our [[WikiJournal of Science/Publishing|guidelines and possible publication formats]], and we have put in place official [[WikiJournal of Science/Bylaws|bylaws]]. [[metawiki:Grants:Project/Rapid/WikiJournal_2018|Funding for 2018]] has been secured (including a small social media advertising budget). ===Proposed priorities for next year=== WikiJMed found that there was a positive feedback loop of having a portfolio of high-quality published articles that leads to submission of other. Therefore here are my priority suggestions for discussion: # Putting out our first issue should be our first priority #* Solicit further article submissions by invitation #* Move current submissions through [[WikiJournal of Science/Potential upcoming articles|peer review pipeline]] # Advertising our existence more broadly #* Posters to put up in departments and institutes (e.g. [[c:File:WJS_Poster.pdf|poster link]]) #* Contact additional scientific societies (e.g. [https://www.isev.org/ ISEV]) #* Soliciting endorsement scientific and open-access groups #* Outreach via [https://www.facebook.com/WikiJSci FB] and [https://twitter.com/WikiJSci Twitter] # Signing up to [http://publicationethics.org/ COPE] ([[WikiJournal of Medicine/Draft of ethics statement|ethics draft]]) # Signing up to [https://doaj.org/ DOAJ] and other indexing services ===Article pipeline=== I hope that we will put out the first issue of peer reviewed articles in the first half 2018. Although we will eventually do continuous publishing, I think that it would be best to simultaneously publish our first set of 4-10 articles together. In addition to the articles currently having their peer review organised, there are at least two submissions expected in January, as well as further tentative expressions of interest. Looking forward to 2018. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:58, 1 January 2018 (UTC) == Manuscript: A card game for Bell's theorem and its loopholes == [https://wikiversity.miraheze.org/wiki/Main_Page '''A card game for Bell's theorem and its loopholes'''] currently resides on Miraheze. I have two suggestions that need to be cleared by my co-author before we can proceed. I think he will approve (when I catch him while he is not busy.) # Move the manuscript into ''[[WikiJournal Preprints]]'' as an '''unsubmitted draft'''. Josh had no objection to placing it on Miraheze, where it was declared as "''copyrighted for submission to a journal''". I see no reason why he would object placing it on [[w:Wikiversity|WV]] provided we don't actually submit to [[WikiJournal_of_Science|WJS]]. # This request is more complicated: ::Joah and I are at opposite ends of our career. I plan to soon devote 100% of my time to [[w:Open educational resources|OER]], using my pension to help fund this effort. I turn 66 in April and don't need a refereed publication. Josh has taken the bold step of offering to publish in WJS, provided '''he can be convinced that the manuscript was properly refereed'''. Josh is not making the more stringent demand that WJS be established in the academic community as a refereed journal (which will probably take years.) When I talk to colleagues on our campus about the WJS, many support the concept but remain skeptical about our chances of success. Josh is willing to gamble on success. He also wants to focus his career on teaching and doesn't need a refereed pub at the moment. Josh agrees with me that a large portion of teaching should be done online, and one of his primary interests is applying concepts of game theory to this effort (which explains his interest in the manuscript). Pending approval from Joah, I will make the following offer: ::'''I have two reviews from a rejection by the American Journal of Physics that I will make available to WJS upon request'''. Neither referee challenged the mathematical accuracy of the article. They did quibble with my sloppy use of the terms "loophole" and "superdeterminism". I tried to make it clear that I was using these words loosely, and am willing to work with an expert who actually knows what they mean. Long ago I published an article on Bell's theorem in the Philosophical Quarterly, and it was reprinted in a collection of essays edited by Theodore Shick.^{[http://www.wright.edu/~guy.vandegrift/shortCV/Papers/bell.pdf ref]} I am certain that this paper is a much better explanation, and reasonably confident that it is mathematically correct.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:48, 15 January 2018 (UTC) :That certainly is a beautifully turned-out piece of work, and to my mind would make a fine WJS article. How do you envision the journal could make use of these existing reviews - supply them as extra material to further reviewers, or use them as actual reviews for this submission? It hadn't occurred to me before to recycle reviews between journals, as it were, but on the face of it there's no reason why not... --[[User:Elmidae|Florian <small>(Elmidae)</small>]] ([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]]) 19:51, 15 January 2018 (UTC) ::At some point in this process, you would need the consent of the "primary" journal because you have no proof except my word that these are actual AJP reviews. --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 20:10, 15 January 2018 (UTC) :::WikiJSci could definitely use the peer reviews organised by other journals with the following requirements: :::*Original journal would need to be contacted in order for editors to find out reviewer identities :::*Reviewers would have to be contacted to ask permission to post peer review comments publicly (even if anonymous) :::Worst case scenario, the peer review could be reorganised from scratch (potentially with the authors nominating the same peer reviewers as they did to the previous journal). Either way, the article would be thoroughly peer reviewed. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 21:13, 17 January 2018 (UTC) ::::Someone should draft a carefully worded letter to AJP about "recycling" those reviews of the Bell's theorem paper.--21:27, 17 January 2018 (UTC) :::::I shall aim to do so in the next week. I shall cc in the two authors of the paper to show the journal that the authors consent to transferring the peer reviews (see [https://authorservices.wiley.com/Reviewers/journal-reviewers/how-to-perform-a-peer-review/general-and-ethical-guidelines.html Wiley guidelines] as example). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 22:01, 17 January 2018 (UTC) ::::::A third (late) review arrived at AJP and the editor forwarded it to me. Like the other two, it found fault with the language used to explain the analysis, but not with its mathematical validity. This review can be seen at https://wikiversity.miraheze.org/wiki/Talk:Main_Page#AJP_Late_report:_Reject_publication_in_present_form --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:02, 18 January 2018 (UTC) :::::::That appears to be admin-protected, can you move it? --[[User:Elmidae|Florian <small>(Elmidae)</small>]] ([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]]) 12:27, 18 January 2018 (UTC) ::::::::{{at|Elmidae}} I will send you an email with instructions for entering the wiki.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:45, 18 January 2018 (UTC) == I mentioned WJS in a message I left on a Wikipedia talk page == Perhaps you should be informed whenever I make an attempt to foster collaboration between WJS and an organization. I will of course inform you of any developments. See [[w:Talk:Del_in_cylindrical_and_spherical_coordinates#I_agree_with_the_above_and_would_like_to_respond]] --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:27, 17 February 2018 (UTC) ::I also linked to WJS on the Miraheze page when I moved the draft into Wikiversity draftspace. Let me know if you want me to do anything differently regarding [[Draft:A card game for Bell's theorem and its loopholes]]. I plan to submit the copyright release as soon as the coauthor the [https://docs.google.com/forms/d/e/1FAIpQLSf-Nu7hjiTeJ5uQ5ozMOIivWZjeyJCPLwAUOuNDP1MVKUbCSQ/viewform Author consent form]. He is now on board with officially submitting it to WJS, which should get us out of '''0''' phase.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:45, 17 February 2018 (UTC) :::{{re|Guy vandegrift}} Thank you for the notice. I've added the {{tlx|article info}} header template for tracking and formatting. I look forward to the submission form. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 04:49, 18 February 2018 (UTC) ==Not knowing the proper venue for author responses to referee comments, I created a subpage== *See [[Draft:A card game for Bell's theorem and its loopholes/Guy vandegrift]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:35, 23 March 2018 (UTC) **Typically, the authors respond on the same page as the reviewers, however that is usually for the 'final version' of a response, as would be submitted to a journal during peer review. This can either be point-by point, or as a new section below the reviewer comments ([[Talk:WikiJournal of Medicine/Plasmodium falciparum erythrocyte membrane protein 1|example]]). Having reviewer comments and author responses collated together on one page can make them easier to track. However you're certainly welcome to use a subpage to draft unstructured / in-progress thoughts. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:39, 24 March 2018 (UTC) ***Yes, this is perfect. I will compose my response in a public place and post it on the talk page. I need to communicate with my coauthor and see what he wants to do. It should not take long.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:09, 24 March 2018 (UTC) == Email now working == The contact email is now working, see [[WikiJournal of Science/Contact]]. I've emailed the board to discuss which people should have access and be responsible for checking emails to this address. [[User:Mikael Häggström|Mikael Häggström]] [[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 12:17, 2 April 2018 (UTC) I guess the email is not working yet. I am trying to contact somebody because my paper is without review since the submission (july, 2021). The editor do not answer my contact, and the editor in cheff too. Is anyone there? [[User:Brunobraga|Bruno Braga]] 21 November 2022. == Review article ideas == Looking around for review articles that would be good for transition to en wiki, I think trying to request [[w:Relaxed selection]] might be a good candidate. It currently redirects to [[w:Evolutionary pressure]] which doesn't even mention it. I found a recent (2009) review of relaxed selection in the wild but could use an update and also a more general treatment (although it also does go over artificial relaxed selection to some extent). Does anyone have any ideas on who to request this from? <ref>{{Cite journal|date=2009-09-01|title=Relaxed selection in the wild|url=https://www.sciencedirect.com/science/article/pii/S0169534709001505|journal=Trends in Ecology & Evolution|language=en|volume=24|issue=9|pages=487–496|doi=10.1016/j.tree.2009.03.010|issn=0169-5347}}</ref> [[User:Mvolz|Mvolz]] ([[User talk:Mvolz|discuss]] • [[Special:Contributions/Mvolz|contribs]]) 12:48, 6 April 2018 (UTC) {{reflist}} == Maximum editorial board size == It's just been brought to my attention that I overlooked a point in the WikiJSci Bylaws that states that "[[WikiJournal of Science/Bylaws#Section 3. Appointment|the number of Editorial Board Members of Wiki.J.Sci. should be kept at a minimum of 10 and a maximum of 20]]" The [[WikiJournal of Science/Editorial board|WikiJSci board]] has just expanded to 25, with two more recent applications [[Talk:WikiJournal of Science/Editorial board|still open]]. It is my error for having overlooked the item, so I apologise for that. We should discuss the options and begin the process of [[WikiJournal of Science/Bylaws#ARTICLE VIII - AMENDMENT|amendment]] (or alternatively close further board additions and let the number fall back to 20 over time). I think that an upper limit of 20 is to small for the journal, but I would be interested in the opinions of others on what the size limits (inf any) should be. My opinion is that the a larger board provides: * A greater scope of knowledge (scientific, publishing and other) * A greater professional network to call upon for invited submissions * A greater group to spread editorial workload over As always, these are only my opinions and I am happy to implement the consensus. Apologies again for the oversight! [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:16, 21 April 2018 (UTC) :Here are my thoughts...as an inactive member. <small>(Thoughts later stricken--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:02, 23 April 2018 (UTC))</small> :# <s>'''A minimum number needs to exist.''' One reason for limiting the membership involves the rare occurrence of an serious and consequential controversy. If it is a difficult decision, then both sides have a valid point. But ultimately the decision comes down to a vote, and those on losing side needs to decide whether to break off and form a new journal or stay. Either decision is justifiable and reasonable. But when the vote is made, only those who are either currently active or highly qualified should participate. </s> :# <s>'''Create two organizations''' One would roughly parallel the open wiki-communities we see on talk pages, perhaps with a structured voting procedure. The other would have an oversight role and make the final decisions. As a person who is neither very active nor highly qualified, I would be happy to join that second organization. In fact, I hate making tough decisions.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 15:23, 21 April 2018 (UTC)</s> :As noted on the email list, there seems to be little downside to a larger-than-expected board size, assuming sufficient discretion is exercised in admitting new members; and more bodies to spread the editorial work over is good. I would suggest that if possible, we amend the statutes to allow for greater board size, and then keep to that limit. --[[User:Elmidae|Florian <small>(Elmidae)</small>]] ([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]]) 13:58, 23 April 2018 (UTC) ::I like [[user:Elmidae|Elmidae]]'s suggestion. With no difficult or divisive controversies on the horizon, we could allow the membership to grow a bit engaging in the difficult task of creating some sort an executive committee. Our highest priority is recruitment of people who can "sell" the idea of a wikijournal to colleagues, and "demoting" members this early seems counterproductive. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]). 16:02, 23 April 2018‎ (UTC) :::Additional note: Discussion on the board mailing list also currently favours an increased or uncapped size of board. A key note is that editorial board members gain access to confidential material (e.g. anon reviewer identities) and so applicant vetting must remain stringent. An additional suggestion was that there could be a yearly confirmation from each board member to confirm that they wish to extend their term. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:42, 24 April 2018 (UTC) ::::I approve of this proposal by [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]. We need to be sure everybody is vetted. The yearly confirmation will help things a bit. Above all, we need to grow in number so this WikiJournal survives and thrives. If our numbers grow too much, we can always diversify our roles. I, for example, would rather be a writer than an editor. For now, we should retain ourselves as a single unit (the WJS editorial board). --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:12, 25 April 2018 (UTC) I do not see the point of having a fixed limit on the board size. A few considerations that could nevertheless limit the board size: # It should be proportionate to the number of submitted articles. # Quality should be as high as possible, i.e. we would like to be in the position of choosing board members among many qualified applicants. # There should be room for actively recruited board members, in addition to people who apply spontaneously. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 20:15, 25 April 2018 (UTC) == FYI: Evidence that the world needs this journal == [https://en.wikipedia.org/w/index.php?title=Del_in_cylindrical_and_spherical_coordinates&curid=753145&diff=839191281&oldid=839018819 '''This edit'''] appeared on my Wikipedia watch list this morning. Advanced physics students can't function without a list of these mathematical identities. The edit shows that Wikipedia's list will never be functional until they find a mechanism to do exactly what the WJS is designed to fix.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 11:34, 2 May 2018 (UTC) == [[Draft:A card game for Bell's theorem and its loopholes]] == I am working hard on this rewrite to make the summer goal for the first edition. If you count the AJP reviews, this is the most thoroughly reviewed paper I ever wrote. Many of the issues raised by reviewers #1 and #2 are discussed in [[Draft:A card game for Bell's theorem and its loopholes/Guy vandegrift|'''this supplement''']]. The ability to publish supplementary material in a wiki side-by-side with the article is a good selling point for WJS. Does the WJS community want such material to be released into WV namespace? Or do you wish to reserve space under WJS for such material? Also, reviewer #3 was extremely meticulous. I have finished responding to their comments on the [[Draft:A card game for Bell's theorem and its loopholes#Abstract|'''Abstract''']] and [[Draft:A card game for Bell's theorem and its loopholes#A_simple_Bell's_theorem_experiment|'''A simple Bell's theorem experiment''']]. I hope to finish in a week or so.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 12:41, 6 May 2018 (UTC) :We can easily publish supplementary material along with the main manuscript (as a Journal/Article/Subpage). We can do any necessary page renames/moves upon publication. I think the only previous time has been a editorial in WikiJMed that had a supplementary table ([[WikiJournal of Medicine/WikiJournal of Medicine, the first Wikipedia-integrated academic journal/Table S1|here]]). It'll be a good way to include the additional detail that is too in-depth for the main text. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:36, 6 May 2018 (UTC) ::That sounds good to me.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 02:22, 7 May 2018 (UTC) :::I think ''"maybe"'' we are done with this. I want to add [[template:Under construction]] to the subpage "Conceptual" and also add a comment to "Tube entanglement" explaining why I felt it was necessary. Then I really '''might''' be {{done}}--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:07, 27 May 2018 (UTC) == Vote: Editorial board size == Although 10 days have now passed (see discussion above), [[https://en.wikiversity.org/wiki/WikiJournal_of_Science/Bylaws#ARTICLE_VIII_-_AMENDMENT|bylaws amendment]] also requires that a formal vote is held about the editorial board size. So, we'll hereby hold a vote here. The result will be the choice with most votes among [[WikiJournal_of_Science/Editors|editorial board members]]. The main alternatives are: *'''Keep the current limit''' of a maximum of 20 editorial board members *'''Increase the limit''' of editorial board members. Please also suggest what limit you'd prefer. Even if this becomes the majority choice, a high discrepancy among suggestions may still show disagreement in the matter. *'''Remove the limit''' Let's have until Sunday, May 13 before reaching a decision. ----------------- *'''Remove the limit''' is my own choice, in conjunction with more restrictive access to passwords and confidential works, so that access is only given to a smaller group of editorial board members and those who are in need of it at the moment. My concern with a large editorial board was mainly a lack of control in these matters, but I now think we can handle it even with a very large board. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 15:56, 6 May 2018 (UTC) *'''Remove the limit''' in conjunction with some compartmentalization, as per Mikael above. [[User:Markus Pössel|Markus Pössel]] ([[User talk:Markus Pössel|discuss]] • [[Special:Contributions/Markus Pössel|contribs]]) 16:08, 6 May 2018 (UTC) *'''Remove the limit''' as per above. At some point we may also test out an [http://guides.library.uwa.edu.au/ld.php?content_id=23218339 open source manuscript management system] (e.g. [https://plos.github.io/ambraproject/Docs-Home.html Ambra], [https://www.manuscriptlink.com/journals manuscriptlink] or [https://pkp.sfu.ca/ojs/ PKP]). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:24, 6 May 2018 (UTC) *'''Remove the limit''', we need the energy but must pay attention to quality and security. [[User:Chiswick Chap|Chiswick Chap]] ([[User talk:Chiswick Chap|discuss]] • [[Special:Contributions/Chiswick Chap|contribs]]) 06:52, 7 May 2018 (UTC) *'''remove the limit'''(not certain if on editorial board questions I can vote, please strike if not, thank you)--[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 11:57, 7 May 2018 (UTC) *'''Remove the limit''' but keep open options for subdivision of members into access and/or responsibility groups. --[[User:Elmidae|Florian <small>(Elmidae)</small>]] ([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]]) 12:18, 7 May 2018 (UTC) *'''Remove the limit''' - Do not hinder the journal's growth. --[[User:Sophivorus|Felipe]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 13:40, 7 May 2018 (UTC) *'''Remove the limit''' - should help the journal grow! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 13:52, 7 May 2018 (UTC) *'''Remove the limit'''--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:59, 7 May 2018 (UTC) *'''Remove the limit''', but introduce stricter rules for accepting new members. (Yes votes minus no votes should be larger than half the size of the board?) And remove inactive members after some time. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:14, 8 May 2018 (UTC) '''Result: Limit removed''' [https://en.wikiversity.org/w/index.php?title=WikiJournal_of_Science%2FBylaws&type=revision&diff=1865990&oldid=1783067]. I can soon look into ways to subdivide access to sensitive material. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:23, 13 May 2018 (UTC) ==Some comments and questions from a university discussion== A recent discussion on a WikiProject prompted an editor to give a presentation about WJS at his university, and he has forwarded to me some comments and questions. I thought it best to throw these open to the group, mostly because I myself am a little foggy about some points (e.g. post-publication editing). I hope the editor will join us here. {{Quote|text=I just gave a presentation about the WikiJournal in our local lab meeting [...]. We had a very intense discussion afterwards, and I thought that might be of interest. First, people (including the professor) are welcoming this the journal as an opportunity to publish OR open access articles for free (open access is considered very important, but the high publication fees of, e.g., PlosOne, can often not be paid). [...] Regarding the publication of OR articles, there were some questions which I could not fully answer as the information provided in the wiki is not that clear: 1) Post publication editing. This hits a nerve, and people argue that they will not be able to regularly track changes made to the article by themselves. According to the guidelines, only authors and editors will be able to do minor changes (which is considered OK), but published articles in the Medicine Journal do not appear to be protected. Based on the discussion, I think that protection of the published pages would be a key feature the journal should offer to gain acceptance, and to underline its determination to be a academic journal. Furthermore, the two different versions (wikitext, PDF), where one can be edited and the other cannot (creating two slightly differing versions) creates confusion. 2) Publication process. One issue is the translation of the manuscript (e.g., word file) to wikitext. Can an author request help with this formatting? When the "confidential" option is chosen, can one submit an ordinary word file, and translate it to wiki text after per review? Even then, the article would need to undergo a short time as a public preprint to complete formatting, so it is not advisable to submit important papers to this journal which need to stay confidential until formal publication? 3) Will it get an Impact Factor in the future? The discussion about the "encyclopedic review papers" was more heated, with the professor disagreeing that this is a good idea. He is of the opinion that Wikipedia and Academia best stay completely separate, as both are completely different things that cannot be combined. Encyclopedic articles are post-academic, and not written for academics but for the general audience. Academic reviews are written by authorities, while encyclopedic reviews are not. Encyclopedic articles are therefore not academic. The Wikipedia approach of constantly evolving articles with many contributors without a single authority behind it has its merits, and is the better way to generate encyclopedic content, the academic journal approach is not suited.}} :I'll keep off point #1. :As regards #2: it is my understanding that we will definitely be able and willing to assist with the wiki formatting. However, I don't think we can handle confidential material in the way described, as public peer review is a central mechanism here. --[[User:Elmidae|Florian <small>(Elmidae)</small>]] ([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]]) 07:39, 23 May 2018 (UTC) ::1) The challenge is to get good contributions, rather than to prevent bad contributions. Allowing anyone to contribute can be useful: any reader can correct a typo. The stable PDF version is here to provide some insurance against unwanted changes, but I would expect the dynamic Wiki version to be better in most cases. ::3) One reason why WikiJournals exist is for Wikipedia articles to count as traditional publications. This means DOI, CrossRef, etc, but I am not sure whether we should go as far as to get an impact factor, a metric that is widely abused and denounced. An impact factor would be particularly meaningless for a multidisciplinary journal. ::4) Combining Wikipedia and Academia is the whole point of WikiJournals. It is good that people who disagree with this point still find that such journals can be useful by providing gratis open access. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 22:15, 23 May 2018 (UTC) :::My opinions on these topics: :::1) I expect that we'll likely eventually protect published papers, but still allow anyone to suggest corrections (similar to [[w:template:Edit semi-protected]]). Currently there is no push to do so because we've never had any questionable edits, but that may change with greater exposure. However I'm certainly keen to retain some mechanism to correct any errors that pass through the review process. :::2) One possibility would be for confidential articles to be fully handled off-wiki, and only converted into wikitext once accepted, with peer reviewer comments also being published at that point. Realistically, the process of converting references in a word file to wikitext is actually the most limiting part. :::3) Impact factors are calculated by [https://clarivate.com/essays/journal-selection-process/ Web of Science]. They have two indices: ''Emerging Sources Citation Index'' (does not assign IF) and ''Science Citation Index Expanded'' (and assigns IF). Being added to either requires "steady publication over 9-month window". Being indexed by WoS is certainly important (along with Scopus and Pubmed). As for IF, although it's a flawed metric, I'd support having it calculated, since it is a draw for authors. However I think the journal will promote itself more on its alternative metrics, like [[WikiJournal_of_Medicine/WikiJournal_of_Medicine,_the_first_Wikipedia-integrated_academic_journal/Table_S1|viewership of materiel integrated into Wikipedia]], and [https://www.altmetric.com/explorer/report/ab3b956e-bf4a-4277-8084-6605d397fcb7 AltMetric scores]. :::4) I actually surprised that encyclopedic reviews would be so controversial. There are plenty of existing examples of review-articles written by academics in such a style. Academic books are often just a bound collection of broad review articles (e.g. [https://doi.org/10.1016/bs.abr.2015.08.005 one of my own]). Indeed there are plenty of niche academic-written encyclopedias that achieve some limited circulation (e.g. [https://www.sciencedirect.com/science/referenceworks/9780122270802 Encyclopedia of genetics]; note the cost ''per article''). I think that WikiJournals will only ever be a minority entry route for content into Wikipedia, but they certainly compare favourably with the [[w:WP:AFC]] process. ::: I'll be interested to hear the thoughts of others. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:43, 23 May 2018 (UTC) ::::Additional note on confidential submissions: I just checked with the Stewards on MetaWiki ([[meta:Stewards%27_noticeboard#Confidential_submission_for_WikiJournals|here]]) and they pointed out that there are several private-access wikis hosted by the WMF for specialised functions, e.g. arbitration committee discussion ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/private.dblist list]). Conceivably, we could some day set up a 'WikiJournal Confidential Submissions' wiki in the same manner. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 00:07, 25 May 2018 (UTC) :::::Regarding 4); there is also [http://www.scholarpedia.org/article/Main_Page Scholarpedia] publishing peer-reviewed encyclopedic review articles, which is widely cited and accepted. The major difference of the WikiJournals appears to be that its review articles tend not to be written by established experts of the field. Therefore I do see the point that the WikiJournal-approach is quite revolutionary, and that reluctance especially from the more conservative scientists can be expected. Maybe this is more of a problem for very specialized review articles than of broader, interdisciplinary ones. To attempt to erase initial reservations, it might be a good idea to discuss these differences and provide some justification on the journal's wiki pages or in the editorial. --[[User:Jens Lallensack|Jens Lallensack]] ([[User talk:Jens Lallensack|discuss]] • [[Special:Contributions/Jens Lallensack|contribs]]) 17:36, 25 May 2018 (UTC) ::::::I think this might be a good idea - either in editorial or home page content form. It may be more of an issue in the eyes of traditionally publishing academics than we realize. As a further illustration, one of the reviewers for the Baryonyx article wrote back to me with a request for more information before he committed to a review, stating {{tq|I want to understand your journal though before providing a full review, as I don’t understand what the manuscript adds to the literature (most reviews papers add insight or opinion) and your journal seems to just be almost a copy of the data already on the Wikipedia page (including the figures).}} (I did my best to expand on it in reply but I fear it wasn't great. He did do the review though :) --[[User:Elmidae|Florian <small>(Elmidae)</small>]] ([[User talk:Elmidae|talk]] · [[Special:contributions/Elmidae|contribs]]) 07:34, 26 May 2018 (UTC) :::::::"add insight or opinion", wow! I'd say, "Spaces in math" contains 10% of insight and 1% of opinion, which is on the wedge of Wikipedia's tolerance. I (like every expert, I guess) would be glad to add more insight or/and opinion, which is why I seek something attached to Wikipedia rather than included into Wikipedia. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|discuss]] • [[Special:Contributions/Tsirel|contribs]]) 08:07, 26 May 2018 (UTC) ::::::::One thing that we can do for insight and opinion is to have a section at the end of a paper dedicated to that. For example the introduction and discussion sections of [[WikiJournal_of_Medicine/Eukaryotic_and_prokaryotic_gene_structure]] were omitted from the Wikipedia page, [[w:Gene_structure|Gene_structure]]. So long as the opinion/perspectives/insights is contained within a specific section of the journal version, it's easy to excise from the Wikipedia version. I just noticed that we completely forgot to mention it anywhere in the [[WikiJournal_of_Science/Publishing|author instructions]]! [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 13:03, 26 May 2018 (UTC) == References don't show on hoverover == It seems like we lack the [[mw:Reference Tooltips|reference tooltip feature]] where if you put your mouse cursor on a reference number, the footnote reference is shown without clicking on the number itself. Not sure if this is because Wikiversity is running on an older version of MediaWiki or because the feature hasn't been rolled out across all projects. Since many major publishers such as Elseiver are implementing similar feature, I think this feature should be enabled pronto. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:28, 30 May 2018 (UTC) :I agree that it'd be a particularly useful feature. I've [https://www.mediawiki.org/wiki/Talk:Reference_Tooltips asked over at MediaWiki] how to activate it here. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:34, 30 May 2018 (UTC) ::Now activated (thanks to the ever-helpful {{u|Dave Braunschweig}}). I've tested it out and it seems to be working well on references and footnotes. Not working when used in author affiliations, but that's not really as important. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:59, 16 June 2018 (UTC) == Indexing by Informit == I've submitted an application to Informit for indexing (same as WikiJMed). They have been very helpful in speeding up how quickly G-scholar founds WikiJMed articles. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:53, 13 June 2018 (UTC) ::[https://search.informit.org/browseJournalTitle;res=IELENG;issn=2470-6345 WikiJSci is now indexed in Informit]. Currently only Issue 1 appears, so I will check whether they only start indexing an issue when it's complete, or if then can index continuously. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:01, 25 September 2018 (UTC) == Application to DOAJ == Application submitted to DOAJ with details largely similar to the [https://doaj.org/toc/2002-4436 WikiJMed entry]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 13:12, 15 June 2018 (UTC) *[https://doaj.org/toc/2002-4436?source=%7B%22query%22%3A%7B%22filtered%22%3A%7B%22filter%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22terms%22%3A%7B%22index.issn.exact%22%3A%5B%222002-4436%22%5D%7D%7D%2C%7B%22term%22%3A%7B%22_type%22%3A%22article%22%7D%7D%5D%7D%7D%2C%22query%22%3A%7B%22match_all%22%3A%7B%7D%7D%7D%7D%2C%22from%22%3A0%2C%22size%22%3A100%7D great example]--[[User:Ozzie10aaaa|Ozzie10aaaa]] ([[User talk:Ozzie10aaaa|discuss]] • [[Special:Contributions/Ozzie10aaaa|contribs]]) 18:40, 18 July 2018 (UTC) ::I checked on progress with DOAJ and their policy is "We will not reply to emails requesting a status update for applications which are less than 6 months old", so I'll email them again in December If we don't hear something sooner. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:33, 26 October 2018 (UTC) :::I've now sent a request for update to DOAJ. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 00:39, 7 December 2018 (UTC) ::::I've sent another request for an update to DOAJ since I've still not heard from them. They were doing some reorganizing at the end of last year, but I'm keen on ensuring that our application didn't get lost in their filing system. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 10:36, 1 May 2019 (UTC) :::::DOAJ got back to me, letting me know that the journal's ISSN was provisional at the time of application, so the application was dropped. I've therefore resubmitted the application. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:38, 7 May 2019 (UTC) ::::::[https://doaj.org/toc/2470-6345 WikiJSci is now listed in DOAJ]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:59, 12 June 2019 (UTC) == WJoS in Wikidata == You may wish to add some or all of these links, to the right hand infobox panel on the front page: * [https://query.wikidata.org/embed.html#SELECT%20%3FWikiJournal_of_Science%20%3FWikiJournal_of_ScienceLabel%20WHERE%20%7B%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22%255BAUTO_LANGUAGE%5D%2Cen%22.%20%7D%0A%20%20%3FWikiJournal_of_Science%20wdt%3AP1433%20wd%3AQ48414191.%0A%7D%0ALIMIT%201000 Wikidata query for articles published by WJoS] * [[:d:Q48414191|Wikidata item about the WJoS]] * [https://tools.wmflabs.org/scholia/venue/Q48414191 WJoS on Scholia] (visualsation of Wikidata data related to WJoS) -- <span class="vcard"><span class="fn">[[User:Pigsonthewing|Andy Mabbett]]</span> (<span class="nickname">Pigsonthewing</span>); [[User talk:Pigsonthewing|Talk to Andy]]; [[Special:Contributions/Pigsonthewing|Andy's edits]]</span> 12:37, 27 June 2018 (UTC) : {{re|Pigsonthewing}} I'd like to add Wikidata links (especially Scolia), but there are articles currently missing ([https://en.wikiversity.org/wiki/WikiJournal_of_Science/Lead:_properties,_history,_and_applications example]). Do you know how often crossref is scraped to update Wikidata or are they typically added manually?. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 04:48, 21 July 2018 (UTC) ::As a relevant note: the [https://tools.wmflabs.org/sourcemd/ sourcemd tool] which adds crossref metadata to wikidata based on a list of dois. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 03:15, 16 February 2019 (UTC) == WJoS articles in Wikidata == I have made some changes to the Wikidata item about [[WikiJournal of Science/Spaces in mathematics]]. The item is [https://www.wikidata.org/wiki/Q55120290 Q55120290] and the changes are in [https://www.wikidata.org/w/index.php?title=Q55120290&diff=703373199&oldid=699820368 this diff]. The most significant change is the addition of the "interwiki" link! Suggestions for further improvements to the data model welcome. <span class="vcard"><span class="fn">[[User:Pigsonthewing|Andy Mabbett]]</span> (<span class="nickname">Pigsonthewing</span>); [[User talk:Pigsonthewing|Talk to Andy]]; [[Special:Contributions/Pigsonthewing|Andy's edits]]</span> 12:48, 27 June 2018 (UTC) :{{re|Pigsonthewing}} Thank you! I think the item structure seems sensible. Is there an automated way to annotate Wikidata from the {{tlx|article info}} template parameters at the top of each article? E.g. a statement on the peer review url, or when multiple linterwiki links are relevant ({{para|w1}},{{para|w2}} etc.). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 05:45, 29 June 2018 (UTC) ::No, and for the relatively small number involved, I doubt one would be developed. However, it would be possible to make the template display the values stored in Wikidata, so that they only need to be crated once; there. <span class="vcard"><span class="fn">[[User:Pigsonthewing|Andy Mabbett]]</span> (<span class="nickname">Pigsonthewing</span>); [[User talk:Pigsonthewing|Talk to Andy]]; [[Special:Contributions/Pigsonthewing|Andy's edits]]</span> 15:12, 30 June 2018 (UTC) :::As a note on this thread for future reference, the [https://tools.wmflabs.org/sourcemd/ SourceMD] tool can be used to batch-create wikidata items for articles and authors from a list of their DOIs and ORCIDs respectively. I've also made a note in the editorial guidelines. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 10:46, 9 October 2018 (UTC) == ORCID == I am Wikimedian in Residence at ORCID; please let me know if I can be of assistance, in that capacity. Don't forget that the template {{Tl|User ORCID}} exists on this wiki (You can see it in use on my user page). Also, [[:en:Wikipedia:ORCID]] may be of interest. <span class="vcard"><span class="fn">[[User:Pigsonthewing|Andy Mabbett]]</span> (<span class="nickname">Pigsonthewing</span>); [[User talk:Pigsonthewing|Talk to Andy]]; [[Special:Contributions/Pigsonthewing|Andy's edits]]</span> 13:59, 27 June 2018 (UTC) :{{re|Pigsonthewing}} Good idea having the identifiers linked where possible. Is there any particular difference between the {{Tl|User ORCID}} and <nowiki>{{</nowiki>[[w:template:Authority control|Authority control]]<nowiki>}}</nowiki> templates? It looks like [[w:template:Authority control|Authority control]] could be a bit more versatile. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:34, 2 July 2018 (UTC) ::{{Ping|Evolution and evolvability}} Late answer, sorry: No, either can be used. <span class="vcard"><span class="fn">[[User:Pigsonthewing|Andy Mabbett]]</span> (<span class="nickname">Pigsonthewing</span>); [[User talk:Pigsonthewing|Talk to Andy]]; [[Special:Contributions/Pigsonthewing|Andy's edits]]</span> 17:35, 30 July 2018 (UTC) == DOI numbers == Coming from Wikipedia, I noticed that the Wikijournal's DOI numbers redirect to the current publicly-editable version of the article instead of the citeable peer-reviewed PDF. For example, https://doi.org/10.15347/wjs/2018.006 links to Radiocarbon dating which has been edited after peer review and could potentially be edited by any member of the public. Shouldn't the DOI link to either the PDF or the specific revision that was accepted? [[User:Dlthewave|Dlthewave]] ([[User talk:Dlthewave|discuss]] • [[Special:Contributions/Dlthewave|contribs]]) 01:24, 9 August 2018 (UTC) :A good point. I think that pointing directly to the PDF is not ideal, since various useful metadata is contained on the webpage. However, the plan is to use [https://www.crossref.org/get-started/crossmark/ CrossMark] to have dois for versions of an article if significant changes are made. At the time, it was considered that the formatting and correction changes were sufficiently minor to not make an official correction through CrossMark, however we could reconsider that. The only difficulty is the additional work involved in [https://www.crossref.org/webDeposit/ submitting an update notice] to crossmark. In other journals, I've observed that crossmark doi updates are rarely used for reference corrections, orcid additions, spelling/punctuation. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 05:54, 9 August 2018 (UTC) == A mechanism is needed for editors to transition to referees == I believe this is going to be a difficult discussion, not because I expect strong disagreement, but because it is a nearly intractable problem. Let me begin on a personal note and leave it to others to move this conversation forward. As coordinator of the [[Wikipedia:Surface Tension]] I felt frustrated by my inability to recruit referees for the manuscript. I believe the broad pedagogical nature of this article is not only source of this difficulty, but also related to my proposed solution: Wikipedia articles are not intended for experts, and for that reason it is not necessary for experts to referee it. I am already sufficiently educated in physics to referee more than half of it, and with a bit of research into measurement technologies, I could probably referee all of it. The same could be said for many Wikipedia articles that the WJS might publish: Any professor of physics, biology, ect., with a broadly based publication record could probably referee most Wikipedia submissions in that field (and if a Wikipedia article cannot be refereed by a PHD with a broad publication record it probably shouldn't be a Wikipedia article!). At the same time, I am having trouble convincing experts in the field that they should serve as referees. My first thought was for me to referee [[Wikipedia:Surface tension]], but after some email discussions with members of this board, I think we all agree that this is going to be a difficult matter. There are a number of mechanisms by which the WJS could fail, but one of them is that we develop a reputation for accepting articles without proper peer review. My initial opinion that we erred in establishing a rule that editors may not referee has morphed into an understanding that we need to be very careful about how an editor can make that transition. A number of questions need to be carefully discussed: #Should there be a transition period between when an editor resigns as editor and begins to serve as a referee? #Should we establish a rule that editors who transition into referee status are prohibited from refereeing anonymously? #At the very least, shouldn't it be disclosed that the refereeing service was performed by someone who was previously an editor? I'm sure there are other questions that need to be posed, and I will wait for others to take over this discussion, especially since it is my intention to make that transition from editor to referee.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:16, 9 August 2018 (UTC) :Per "should we accord the same "sign-off authority" to self-appointed reviewers as to invited ones? It's not a combination I have seen before, so I don't know what precedents exist." This is interesting! When deciding which journal to submit an ms to, I usually scrutinized the Editorial board because that was where many of the anonymous reviews came from! Other invited reviews also occurred. With the change to open reviewing, it's a much more refreshing environment, I believe. :Regarding self-appointed reviewers, my comment above describes earlier Editorial boards. In the beginnings of the WikiJournal of Science most reviews were performed by the Editorial board, as I recall. I only review where I have some applicable expertise. One point added to the Editorial board guidelines is that board members do not perform reviews but only invite outside peer reviewers. The "Lead" ms arrived on 22 November 2017. Courtesy requires a review within a month. We could not do that so I began "Editorial comments" on 7 December 2017. :The submission [[WikiJournal Preprints/Surface tension]] was received on 20 June 2018 and editorial comments and changes began on 24 July 2018, which was good! The first outside peer review was received on 9 August 2018, excellent! :I have applicable expertise regarding surfaces of solids before and during sputtering where surface tension enters in and was happy to read in our first outside open peer review "surface tension is not limited to fluids but is found in solids as well." --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:59, 10 August 2018 (UTC) :: My answers to [[User:Guy vandegrift|Guy vandegrift]]'s three questions: ::# No transition, no need to resign either. ::# Very good idea: editors who act as referees should not be anonymous, and should explicitly declare that they are editors. COIs are not necessarily avoidable or even bad in themselves, it is <u>undisclosed</u> COIs that should be avoided. (Still, I do not see the COI in this case.) ::# As above. ::[[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 12:31, 11 August 2018 (UTC) :::I looked at the one referee report on "Lead" and it seems very sophisticated (so high-level that I cannot immediately judge it). So for now, the is no immediate need to resign and become a referee (instead I can try and sort out what the referee said.) Regarding [[User:Sylvain Ribault|Sylvain Ribault]]'s suggestion that editors can act as referees, I am <u>personally</u> inclined to agree with them. But I do not consider that personal inclination to be adequate. This a new journal, and we need to careful with our reputation. If in the future we do change the rules to allow editors to referee, it must be after a long and deliberate discussion, and only if we can establish a need for this change. Fortunately, the referee who stepped up to do "Lead" may have given much need time to carefully deliberate the referee/editor question. I do think the WJS role as a "referee" for broadly-based Wikipedia articles might justify a slightly different policy in this regard. :::For now, our number one goal is to recruit scholars to actively support this new journal.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:04, 12 August 2018 (UTC) ::::As a note for external readers, there are a few examples of where editors have provided peer review comments (clearly marked as from editor): ::::*[[Talk:WikiJournal of Science/Radiocarbon dating#Editorial_comments|Example 1]] (Radiocarbon dating) - detailed feedback from specialist editor ::::*[[Talk:WikiJournal of Science/Lead: properties, history, and applications#Editorial_comments|Example 2]] (Lead) - detailed feedback from specialist editor ::::*[[Talk:WikiJournal_of_Science/Spaces_in_mathematics#Commentary_by_non-specialist_editor|Example 3]] (Spaces in mathematics) - clarity recommendations from non-specialist editor ::::*[[Talk:WikiJournal_of_Medicine/Plasmodium_falciparum_erythrocyte_membrane_protein_1|Example 4]] (WikiJMed PfEMP1 protein) - minor recommendations from several editors ::::However in each of these cases, there were also at least two other external peer reviewers. An editorial board member may organise external reviewers for a great many articles, whereas external reviewers very rarely review more than one article. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:21, 12 August 2018 (UTC) :::::I think the policy of requiring two referees is approximately correct. There is certainly no need to tinker with it in the near future--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:29, 13 August 2018 (UTC) == Anticipating an article without using Wikipedia content == I appreciate the (re-)publication and modification of Wikipedia articles. However, I also anticipate an article that doesn't copy or rely too much on Wikipedia. I'll put this another way: when will I see an article that is researched and written in an original way and not copied from Wikipedia? --[[User:George Ho|George Ho]] ([[User talk:George Ho|discuss]] • [[Special:Contributions/George Ho|contribs]]) 22:01, 24 August 2018 (UTC) :If the WJS gets sufficient quality and quantity in the non-Wikipedia category the WJS should consider focusing on non-Wikipedia articles on the grounds that another wiki-journal could be created that specializes in Wikipedia articles. For that reason, we certainly welcome non-Wikipedia articles. At least, that is how I look at it.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:59, 24 August 2018 (UTC) ::We have a few examples so far. Both the [[WikiJournal of Science/Lysine: biosynthesis, catabolism and roles|Lysine]] and [[WikiJournal of Science/ShK toxin: history, structure and therapeutic applications for autoimmune diseases|ShK toxin]] were not adapted from existing Wikipedia content, but instead written from scratch. Both were improvements over the previous Wikipedia pages so were used to subsequently overhaul them. The [[WikiJournal of Science/A card game for Bell's theorem and its loopholes|Bell's Theorem card game]] article is different in that it was not intended to replace a Wikipedia page, but to supplement it as a teaching tool. Over in WikiJMed, there are also examples of [[WikiJournal of Medicine/Acute gastrointestinal bleeding from a chronic cause: a teaching case report|case studies]] (again, with a teaching focus), [[WikiJournal of Medicine/Vitamin D as an adjunct for acute community-acquired pneumonia among infants and children: systematic review and meta-analysis|original metaanalyses]], and topics that were [[WikiJournal of Medicine/Plasmodium falciparum erythrocyte membrane protein 1|previously absent from Wikipedia]]. My strong hope is that as the journals grow, they will attract more and more non-wikipedians to contribute, as a supplement to the traditional wikipedian editor core. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:13, 25 August 2018 (UTC) :::Maybe after a time I'll try "[[b:User:Tsirel/sandbox|Can each number be specified by a finite text?]]" (unfinished draft for now). [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|discuss]] • [[Special:Contributions/Tsirel|contribs]]) 04:58, 25 August 2018 (UTC) == Declined articles: what happens to reviews if the draft is deleted? == If a submission from Wikipedia is declined after receiving substantial reviews, the [https://en.wikiversity.org/wiki/WikiJournal_of_Science/Editorial_guidelines#Article_amendments_and_publication_decision editorial guidelines] recommend that a notice be put on the Wikipedia talk page pointing to the reviews. However, the authors have the option of requesting that the draft be deleted: what happens to the reviews then, and to the link from the Wikipedia talk page? It makes little sense to delete the draft and keep the reviews, as the reviews are less understandable without the draft, even if the draft differs very little from the Wikipedia article. (WikiJSci drafts have figure numbers, for example.) I propose that drafts that have been peer reviewed can never be deleted. An additional benefit would be that the draft could more easily be amended and resubmitted later. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:49, 9 September 2018 (UTC) :I tend to agree. In the early days of the journal we received these submissions: [[Demostration of the No Relativity of Time]] and [[Irrefutable Truths of Life]] which have been deleted but I can restore them if there is consensus. [[Draft talk:Demonstration of the No Relativity of Time]] and [[Talk:Life/Life and Love!]] still exist. It is unlikely that the author could more easily amend and resubmit later as he was blocked by Braunschweig but others might. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 20:32, 9 September 2018 (UTC) :: Although it initially struck me as counter-intuitive, I think you're probably right that it'd be best to keep declined drafts online. Preprint servers like arXiv and bioarXiv do something very similar and I think it's sensible to default to the precedent unless we have strong reasons to do something different ([https://arxiv.org/help/withdraw arXiv guidelines]). If there is consensus I can edit the {{tlx|Article info}} template accordingly. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 00:48, 10 September 2018 (UTC) ::: In general I support keeping reviewed drafts. The only exception that occurs to me is if a draft is frankly absurd or is being used as a vehicle for something forbidden, such as libel, a political campaign, advertising, or hate speech. I doubt if any such thing would normally get as far as being formally reviewed, but I suppose a Trojan horse style attack (exposing itself only at a late stage) is possible; such a thing would then have to be deleted. [[User:Chiswick Chap|Chiswick Chap]] ([[User talk:Chiswick Chap|discuss]] • [[Special:Contributions/Chiswick Chap|contribs]]) 07:07, 10 September 2018 (UTC) :: [[User:Marshallsumter|Marshallsumter]] raises a good point about the retroactivity of the new rule if it is adopted. I find it strange that talk pages have been kept while drafts have been deleted: the former makes little sense without the latter. In the two cited examples the talk pages could be deleted as well, as there was apparently no formal external peer review, and the submissions were not serious. :: If we keep declined submissions they should be listed somewhere, possibly in [https://en.wikiversity.org/wiki/WikiJournal_Preprints Preprints]. If their number increases the issues of numbering, searching, indexing, etc, might arise. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 20:53, 11 September 2018 (UTC) :::One compromise between keeping or deleting certain articles might be [https://en.wikiversity.org/wiki/Special:PrefixIndex?prefix=&namespace=118 Draft space] --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:51, 13 September 2018 (UTC) ::::Good point. I propose that we add on step to the editorial process, where the editors decide whether the submission deserves to be sent to external peer reviewers. In the case of preprints that are not submitted to the journal, we could also do a basic sanity check analogous to arXiv's. Submissions that fail at this stage can be kept in Draft space. Submissions that pass can be treated as preprints, permanently kept, and listed somewhere. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:23, 13 September 2018 (UTC) :::::I have found this: [[Draft:Demonstration of the No Relativity of Time]] which has a deletion tag on it. Apparently, submissions that may not deserve to be sent to external peer reviewers that end up in Draft: ns can be deleted. But, should they be? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:44, 19 September 2018 (UTC) It seems that we have consensus. I have changed the {{tlx|Article info}} template, removing the suggestion that authors can request deletion of declined articles. As far as I can see there is no text (in the [https://en.wikiversity.org/wiki/WikiJournal_of_Science/Ethics_statement#Data_retention_policy ethics statement] or elsewhere) that commits us to offering this option to authors, and the CC-BY license probably allows us to keep submitted articles. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:19, 18 September 2018 (UTC) The [[Template:WikiJPre declined]] contains an option for the author(s) to request deletion. From the above it appears this option should be removed. Agreed? --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 01:25, 24 September 2018 (UTC) :Very much agreed, I just did it. Thanks for pointing it out. [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:10, 24 September 2018 (UTC) ::Good work on this. Another example of precedent is that rejected [http://topicpageswiki.plos.org/wiki/Main_Page PLOS Topic Pages] are also retained on their website in perpetuity. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:12, 26 October 2018 (UTC) {{outdent}} Discussions such as this should occur at [[Wikiversity:Deletions]] or a community wide review. These issues require wider consensus than merely from participants in a WV subproject. The request to undelete these pages is declined. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 14:17, 8 January 2020 (UTC) :(Thanks mikeu for centralising the discussion back here. I accidentally forked over to [[Talk:WikiJournal_User_Group#Under_what_circumstances_should_articles_be_deleted|a separate talkpage]]). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 04:35, 9 January 2020 (UTC) :I'd like to follow up on the rationale for the statements that I made here. On an open wiki where anyone can edit it is likely that we are going to get frivolous or even malicious (vandalism) "submissions." I'm using [[w:scare quotes]] here as I don't consider the act of merely saving a page to the journal to be a true act of submission. There should be a first level of vetting before a page is considered to be a legitimate submission. Otherwise you are just going to permanently preserve My Pet Theory That Einstein Was Wrong (at best) or worse People Of The Foo Race Are Intellectually Inferior. Hand-wringing about the sanctity of pre-print submissions is all fine and well, but you must admit that there are likely numerous submissions to print journals that wind up in the paper shredder with no record left of there existence? In any case deleting vandalism or contributions by an indef blocked user go beyond the scope of any particular project and are subject to community wide decision making. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:37, 13 January 2020 (UTC) ::Since this topic has re-surfaced via a [[Talk:WikiJournal Preprints/COVID-19 ELIMINATION AND CELL DIFFERENTIATION|recent article]], I wanted to put some more general thought in order. ::*In all cases, I'd be very keen to at least keep the metadata + abstract (+ reviews) for transparency of the record. ::*By default, I'd also be keen to keep the content of the article up for both reviewed and unreviewed preprints (this is the norm for other preprint servers like the arXivs, OSF, and preprints.org). ::**The exception is if there are obvious reasons to delete (copyvio, defamation, hatespeech etc). ::**The difficulty is when the author prefers deletion. So far I don't think there's evidence that disallowing deletion would discourage submissions, but it's a possibility. ::*I can imagine authors requesting contact info or name to be redacted. Although I don't know of particular precedent for that), but it's not a million miles from the double-blind ethos in many humanities & soc sciences journals. :: [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 13:05, 23 June 2020 (UTC) I've had some preprints of my submissions accidentally published without my consent at another journal. That popped up in another journal who used it as a basis to decline my work. Bottom-line: I think authors should have a right to have their submission deleted upon request. Otherwise we keep it up. --[[User:Stevenfruitsmaak|Steven Fruitsmaak]] <small>([[User_talk:Stevenfruitsmaak|Reply]])</small> 19:27, 24 June 2020 (UTC) == Dedicated email address for submissions? == The [https://en.wikiversity.org/wiki/WikiJournal_of_Science/Editorial_guidelines#Receiving_a_submission Editorial guidelines] state that 'the corresponding author may write the article online or email it to Submissions@WikiJMed.org'. Should we just delete this, or create an analogous email address for WikiJSci? [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:25, 13 September 2018 (UTC) :I think we can just use Contact@WikiJSci.org for this purpose. For now, there's no real difference between a general submission and contact email (indeed, the contact address has had only four spontaneous presubmission inquiries since its creation, so is not heavily trafficked yet). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:38, 26 October 2018 (UTC) ::OK, sounds good. Who reads the emails to Contact@WikiJSci.org? [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 18:53, 26 October 2018 (UTC) :::Currently all emails to it are automatically forwarded to the EiC (me) and. All board members can check it via the log-in details [https://groups.google.com/forum/?utm_medium=email&utm_source=footer#!forum/wjsboard here]. It would be a good idea to have a set of board members who also volunteer to have its emails forwarded to them. I'll send an email around the board in the near future. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:37, 26 October 2018 (UTC) ::::OK. These logins and passwords appear to be publicly visible, is this a feature or a bug? [[User:Sylvain Ribault|Sylvain Ribault]] ([[User talk:Sylvain Ribault|discuss]] • [[Special:Contributions/Sylvain Ribault|contribs]]) 19:53, 28 October 2018 (UTC) :::::They should only be visible to those on the editorial board (ie. members of WJSboard@googlegroups.com) so that e.g. any board member can use the crossref login to mint new DOIs. For others (or when not logged in) it should display "You must be signed in as a member of this group to view and participate in it". [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:24, 29 October 2018 (UTC) == WJoS and Google Scholar == A quick learned that articles can be found in Google Scholar. But there is no item/link in the right column to the full text like Researchgate has/does. I guess people are more likely to click on a link in the right column to get to the full-text. Jacques Versteijnen 4 oct 2018 :Yes, we've had problems with Google Scholar being able to correctly index articles. For some it finds the full text versions ([https://scholar.google.co.uk/scholar?cluster=13408144424759452960&hl=en&as_sdt=0,5 example]), but for others, it seems to only find the informit indexed abstract ([https://scholar.google.co.uk/scholar?hl=en&as_sdt=0%2C5&q=wikijournal+of+science+ShK+toxin%3A+history%2C+structure+and+therapeutic+applications+for+autoimmune+diseases&btnG= example]). I've contacted Google Scholar twice about this since June, however they've traditionally had a low contact response rate. I'll update this location if I hear back. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 00:04, 27 October 2018 (UTC) == Transpose policy database == Having only just been made aware of the [https://transpose-publishing.github.io/ Transpose] open database of journal policies, I have added URL links to them for WikiJSci, along with a note that they are the same for WikiJMed and WikiJHum. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:06, 19 January 2019 (UTC) :WikiJSci is now listed in the '''[https://asapbio.org/transpose-preprints TRANSPOSE database]'''. I shall check to see what the process is on adding the other journals. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 22:45, 20 March 2019 (UTC) == WJoS with Academic Libraries == [[File:AIT Library - WikiJournal of Science - 1.jpg|thumb|WikiJSci poster in [https://en.wikipedia.org/wiki/Asian_Institute_of_Technology Asian Institute of Technology] Library window]] In an attempt to increase the visibility of [https://en.wikiversity.org/wiki/WikiJournal_of_Science WikiJournal of Science] and to have more publishing contributions from Academic students and faculty, it is more important to connect academic libraries. :[https://en.wikipedia.org/wiki/Asian_Institute_of_Technology Asian Institute of Technology] Library has shown consent in promoting [https://en.wikiversity.org/wiki/WikiJournal_of_Science WikiJournal of Science] in its activities and encouraging Students and Faculty members towards [https://en.wikiversity.org/wiki/WikiJournal_of_Science WikiJournal of Science]. ::[https://en.wikiversity.org/wiki/WikiJournal_of_Science WikiJournal of Science] Poster Display on - Academic Library Screens/outdoor signage([https://commons.wikimedia.org/wiki/File:AIT_Library_-_WikiJournal_of_Science_-_1.jpg Image1], [https://commons.wikimedia.org/wiki/File:AIT_Library_-_Wiki_Journal_of_Science.jpg Image2] ). | --[[User:Gorlapraveen123|Gorlapraveen123]] ([[User talk:Gorlapraveen123|discuss]] • [[Special:Contributions/Gorlapraveen123|contribs]]) 21:45, 8 March 2019 (UTC) :::Excellent, thank you! They look particularly good on a screen like that. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 05:17, 9 March 2019 (UTC) == title typo? == The recent inclusion has a header of "VOLUME 1 (2019) ISSUE 1" Should that be vol. 2? --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 13:54, 11 March 2019 (UTC) :Thanks for the note! The typo has been corrected. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:47, 11 March 2019 (UTC) == A recent 10-fold increase in page views. Does anybody know why? == I keep track of page views to one WJS article and noticed a sudden increase. It seems to be associated with a sudden increase traffic for the WJS. Does anybody know why? See <br>https://tools.wmflabs.org/pageviews/?project=en.wikiversity.org&platform=all-access&agent=user&range=latest-90&pages=WikiJournal_of_Science<br> It started on 3/18/2019. --[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:50, 22 March 2019 (UTC) :I suspect it's the recent publication of the [[WikiJournal_of_Science/RIG-I_like_receptors|RIG-I like receptors]] article based on comparing the [https://tools.wmflabs.org/pageviews/?project=en.wikiversity.org&platform=all-access&agent=user&start=2019-01-01&end=2019-03-21&pages=WikiJournal_of_Science|WikiJournal_of_Medicine|WikiJournal_of_Humanities|WikiJournal_of_Science/RIG-I_like_receptors|WikiJournal_of_Science/Peripatric_speciation|WikiJournal_of_Science/Lead:_properties,_history,_and_applications pageviews for it and for the main page]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:57, 22 March 2019 (UTC) ::On the logarithmic scale the effect is [https://tools.wmflabs.org/pageviews/?project=en.wikiversity.org&platform=all-access&agent=user&start=2019-01-01&end=2019-03-21&pages=WikiJournal_of_Science|WikiJournal_of_Science/RIG-I_like_receptors|WikiJournal_of_Science/Spaces_in_mathematics|WikiJournal_Preprints/Can_each_number_be_specified_by_a_finite_text%3F visible] also for less visited articles and even preprints. &nbsp; :-) &nbsp; [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|discuss]] • [[Special:Contributions/Tsirel|contribs]]) 05:24, 22 March 2019 (UTC) == Moved preprint for "Ice drilling" == I just moved the preprint for "Ice drilling" to "Ice drilling technology", based on feedback from one of the reviewers, but it appears to have broken some internal organization that I don't understand. Would someone be kind enough to fix things up? Thanks. [[User:Mike Christie|Mike Christie]] ([[User talk:Mike Christie|discuss]] • [[Special:Contributions/Mike Christie|contribs]]) 21:22, 23 March 2019 (UTC) :You moved the talk page but not the article page. If you're sure you want to change the article title just move [[WikiJournal Preprints/Ice drilling]] to [[WikiJournal Preprints/Ice drilling technology]]. It's your call. Google books has "Ice drilling" in the title of 6,080 books, including "Ice drilling technology", but "Ice drilling technology" occurs in 6,410 books. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 22:17, 23 March 2019 (UTC) ::Also, on Google books: "Ice drilling" -technology, is about 3,550 results so ice drilling in some form appears more often than with "technology". --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:28, 24 March 2019 (UTC) :::I've moved it; thanks. I think the reviewer feedback made it clear to me that the article is really only about the technology, not about logistics, grants, or any of the related activities. [[User:Mike Christie|Mike Christie]] ([[User talk:Mike Christie|discuss]] • [[Special:Contributions/Mike Christie|contribs]]) 21:56, 26 March 2019 (UTC) {{:WikiJournal_of_Science/Applications/SCOPUS}} == SHERPA/RoMEO == I've submitted to the details for WikiJSci to SHERPA/RoMEO via the [http://sherpa.ac.uk/forms/new-journal.php journal submission form]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:50, 2 May 2019 (UTC) == Some questions == Hi everyone here and congratulations on the idea of developing a journal. At Wikispecies where I am a Beurocrat and CheckUser I noticed a post about this site [https://species.wikimedia.org/wiki/Wikispecies:Village_Pump#A_proposal_for_WikiJournals_to_become_a_new_sister_project here]. In anycase although I have no issue with developing that proposal I have a couple of questions about the journal. A journal when publishing nomenclatural acts must comply with the ICZN code for Zoology. So this means publishing new species names, but also moving species to new genera, ie combinations, and any other act that effects the usage of names in zoology. If you have no plan to ever publish nomenclatural acts this should be in your authors guidelines. Otherwise your journal needs to meet the code if you ever do publish one. For example electronic journals must publicly state their archiving agency and further this must be stated in the Zoobank registration of the journal, which will include the LSID of this registration being published in the journal. The date of publication is the date upon which the paper meets all requirements of the code and at this point no further changes can be made. As in once published a nomenclatural paper cannot be altered, or it becomes unavailable. I also am interested in how you would get a nomenclatural paper reviewed if you ever receive one. Anyway I am happy to help with the journal if possible, I can be found on Wikispecies if needed. Am also now watching this page. Cheers [[User:Faendalimas|Faendalimas]] ([[User talk:Faendalimas|discuss]] • [[Special:Contributions/Faendalimas|contribs]]) 15:58, 9 June 2019 (UTC) :Welcome to the WikiJournal of Science! While the need to comply with the ICZN code for Zoology is not immediate, having edited Wikispecies occasionally and read a small number of articles regarding nomenclatural acts, I can foresee the likelihood of such a submission perhaps from Wikispecies contributor(s). We also have several biologists and zoologists who may be interested in finding qualified reviewers on our editorial board. Our EiC is awesome at bringing our journal inline with various publishing standards and may be interested in this. Thank you for bringing this to our attention! If such acts are within your areas of expertise, please consider applying to the board. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 22:24, 9 June 2019 (UTC) ::I would be happy to apply to the editorial board if you think that would help. Yes nomenclatural taxonomy is an area of my expertise and I have published on this fairly extensively. I am also an editor of Zoobank so can help with the registration process if its ever needed. I am also happy to be a reviewer if needed. My editorial eperience in Wikimedia is largely on WP and Wikispecies. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 15:12, 10 June 2019 (UTC) :::{{re|Faendalimas}} I absolutely agree with putting together some guidelines for nomenclature articles. The relevant parts of the [https://www.iczn.org/the-code/the-international-code-of-zoological-nomenclature/the-code-online/ ICZN publication requirements] look as though they should be implementable. I'd be very interested in your perspectives on this, since I'm keen to avoid any possibility of the journal being used for [https://blogs.scientificamerican.com/tetrapod-zoology/taxonomic-vandalism-and-hoser/ taxonomic vandalism]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:38, 10 June 2019 (UTC) ::::Sure thing, I can discuss these things here or we can get on IRC or something up to you all. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 00:16, 11 June 2019 (UTC) :::::As part of addressing this and expanding the guidelines in general (per comment 102 on [[metawiki:WikiJournal#Discussion|this page]]), I've created a draft page here: [[WikiJournal User Group/Guidelines/Draft]]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:14, 28 June 2019 (UTC) ::::::{{re|Faendalimas}} A [[WikiJournal User Group/Guidelines/Draft|first draft]] of improved author guidelines is ready. Would you be interested in helping write 50 word ICZN code summary for that page and a longer set of ICZN guidelines in a separate page that can be linked to? [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 05:14, 22 July 2019 (UTC) == problematic promises and potential conflict of interests == In the orientation for authors there is currently the following line: :''Articles that pass peer-review are published as a citeable, indexed PDF, and suitable text and images are integrated into Wikipedia and related projects (with a link to the indexed PDF). The vast readership of Wikipedia results in a high effective impact of included works.'' I think in this form that "promise" is rather problematic. First of all whether content from a WikiJournal article is included into WP articles or whether it gets linked there is not up to WikiJournal (its editors or authors) but it is up to the the WP community in general and in particular up to the editors of the concerned WP articles and the WP projects maintaining them. Secondly it is not really clear to me why WikiJournal articles should receive preferential treatment over other Open-Access journals or other other reputable external sources in general. I think that formulation should be modified to to make clear that texts and images ''may'' be used in WP (''at the discretion of WP editors'').--[[User:Kmhkmh|Kmhkmh]] ([[User talk:Kmhkmh|discuss]] • [[Special:Contributions/Kmhkmh|contribs]]) 01:06, 10 June 2019 (UTC) :I agree with what you are stating! The key word in the orientation is "suitable", so far this has been decided by the WP editors involved with the Wikipedia GA, FA, or developing article submitted for possible publication in the WikiJournals and I believe also involves WP WikiProject participants. Neither the WikiJournal Editorial Boards nor the professional reviewers are focused on Wikipedia, but are focused on the professional standing of the WikiJournals, which also publish original research that can be included in a submission after import of the Wikipedia article. Of course the OR does not go into the WP article, for example. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:49, 10 June 2019 (UTC) ::If I a understanding this correctly, no journal should ever be receiving what could be perceived as preferential treatment. This goes against the principal of academic freedom. You want people to utilise the journal, of course, but you want this to be because they wish to use it because of the science presented. Not for any other reason. If a journal strives for quality it will be used. Cheers [[User:Faendalimas|<span style="color: #004730">Scott Thomson</span>]] (<small class="nickname">Faendalimas</small>) ^{[[User talk:Faendalimas|<span style="color: maroon">talk</span>]]} 17:08, 10 June 2019 (UTC) :::{{re|Kmhkmh}} I agree that the journals shouldn't aim for any kind of override of community consensus on any Wikipedia article (and should definitely avoid any edit war). . The intention is to write something that is also clear to non-wikipedian potential contributors, whilst also being accurate (We also have a note about this in the [[WikiJournal User Group/Ethics statement#Wiki ownership|ethics statement]]). Aiming for specificity as well as concision, perhaps "...integrated into Wikipedia (and related projects) by consensus of the editor community" or "...in discussion and with consent the editor community". Although we've not had any problems so far, it's good to think ahead. I'd also want to avoid someone having false expectations of e.g. submitting a review of RNA interference and expecting that it would definitely wholly replace the [[W:RNA interference|current Wikipedia page]]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 02:30, 11 June 2019 (UTC) ::::I've added a note at this point [[WikiJournal User Group/Publishing#cite note-WP-1|here]] and [[WikiJournal User Group/Editorial guidelines#Wikipedia inclusion|here]] for now. More to follow as part of updating the [[WikiJournal_User_Group/Publishing#General_guidelines|General guidelines]]. I'd be interested in suggestions for improved wording. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:50, 26 June 2019 (UTC) == Original research == The text in the box for “original research” refers to <q>Such papers follow the standard ''Introduction, Results, Discussion, Methods'' format,…</q> but other style docs refers to IMRaD as the more common list of sections, that is ''Introduction, Method, Results, Discussion'' format. [[User:Jeblad|Jeblad]] ([[User talk:Jeblad|discuss]] • [[Special:Contributions/Jeblad|contribs]]) 15:22, 11 July 2019 (UTC) == Unable to post in the wjsboard Google group == I would like to reply to [https://groups.google.com/g/wjsboard/c/y6mmRRf26PM/m/iE4VaVNrBwAJ this thread] in the wjsboard Google group about [[WikiJournal_Preprints/Speciation_by_reinforcement|the paper ''Speciation_by_reinforcement'']] for which I'm the assigned editor (Emanuele Natale), but I get the message ''You do not have permission to post to this group''. Perhaps my account was not properly added to the group? --[[User:Natematic|Natematic]] ([[User talk:Natematic|discuss]] • [[Special:Contributions/Natematic|contribs]]) 17:37, 25 March 2021 (UTC) :{{re|Natematic}} I've gone in to check the settings and I ''think'' I've found the issue. Could you try again now? [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 22:54, 25 March 2021 (UTC) :: @[[User:Evolution_and_evolvability|T.Shafee]] thanks, you resolved the issue: I confirm that now I'm able to reply. [[User:Natematic|Natematic]] ([[User talk:Natematic|discuss]] • [[Special:Contributions/Natematic|contribs]]) 08:57, 29 March 2021 (UTC) == Special Issue on Open Science Fellows Program == I was wondering if the WikiJournal of Science has a procedure to handle special issues? In the context of the [[Wikimedia Deutschland/Open Science Fellows Program|Open Science Fellows Program,]] we are planning a special issue that reflects the program. I think the WikiJournal publishing concept would be best suited for the spirit of this program among all publishing concepts I am aware of. Any feedback / response would be highly appreciated. {{ping|Daniel_Mietchen}} {{ping|Evolution_and_evolvability}} [[User:Physikerwelt|Physikerwelt]] ([[User talk:Physikerwelt|discuss]] • [[Special:Contributions/Physikerwelt|contribs]]) 06:45, 24 October 2021 (UTC) :Back in October 2021, we were discussing with an Ukrainian editor for a possible special issue from Ukrainian community. It is now on pause due to the ongoing war. I think special issue can be explored further. Happy to discuss more. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:23, 29 March 2022 (UTC) :{{re|Physikerwelt}} Now that we have cleared most of our backlog, we can resume the consideration of special issues. What kind of publications do you have in mind? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:23, 14 April 2023 (UTC) == Recruiting technical editors == We are hiring new [[WikiJournal User Group/Technical editors|technical editors]] for the journals. Please see [https://www.linkedin.com/posts/andrewcleung_technical-editor-job-poster-activity-6912636772371828736-LteF this job posting for details.] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:23, 29 March 2022 (UTC) == How to become a member of the editorial board or reviewer. == I want to become a member of the editorial board or reviewer. [[User:HariSinghw|HariSinghw]] ([[User talk:HariSinghw|discuss]] • [[Special:Contributions/HariSinghw|contribs]]) 04:05, 6 July 2022 (UTC) :{{re|HariSinghw}} There are links to apply at [[Talk:WikiJournal of Science/Editors]]. Essentially it involves 100-200 words to describe your previous relevant professional, editorial, and open access experience. Alternatively, you can do the same thing via a google form [https://docs.google.com/forms/d/e/1FAIpQLScM6WliEQOUWQxBrCsUHKV5ZeeIuEwfTqrZDieztGIFbgfKdQ/viewform?usp=sf_link here]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 01:26, 7 July 2022 (UTC) == Proposal to introduce "Inactivity removal policy" to the bylaws == There is an ongoing discussion to propose introducing an inactivity removal policy for editorial board members. Full details [[Talk:WikiJournal User Group#Proposal to introduce "Inactivity removal policy" to the bylaws|can be viewed here]]. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:24, 12 September 2022 (UTC) == Editor in Chief position == Since I’m taking on the Exec Director role for the WikiJournal User Group, I’ll be stepping down from the Editor in Chief position to maintain separation between editorial and executive decision-making. I’ll of course keep helping out supporting WikiJSci, but making space for others to take leadership over the journal! It’s been great being the EiC and I look forward to continuing to support the journal. ===Process=== The [[WikiJournal of Science/Bylaws#ARTICLE IV - EDITOR-IN-CHIEF|relevant section of the bylaws]] states that: :{{tq|The ''Editor-in-chief'' is appointed by consensus in the ''Editorial Board''.}} So, I recommend that anyone with an interest in the position reply below before to 21^st of April with a paragraph or so on what they’d bring to the role. We can then establish consensus over the following week and formally hand over the role at the end of that. Technically only the consensus of current board members is considered, but those not on the board are welcome to comment/discuss. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:41, 6 April 2023 (UTC) :I would like to put my name forward for consideration for the editor-in-chief position. I have been the managing editor for WikiJournal of Science for over a year, responsible for vetting submissions and arranging peer reviews for publications. Through these submissions, I have demonstrated my ability to consider different opinions of editorial board members and make a decision on publishing or declining a submission. I have also assisted in the recruiting and onboarding of new technical editors and associate editors, including those from Medicine and Humanities, by giving orientation presentations and answering their questions. In the monthly editorial board meetings, I frequently contributed to the discussion agenda and offered my thoughts on the subject matter. I believe I have the experience to further advance the work of WikiJournal of Science. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:47, 21 April 2023 (UTC) Since Andrew's Expression of Interest is uncontested and I've complete confidence in his suitability for the position, I have gone through and implemented the change. Congratulations Andrew! I'm very happy to announce that I'm handing over the role of Editor in Chief over to Andrew Leung. He's been a huge asset to WikiJSci ever since he joined the journal in 2018. He's very skilled at working across the broad range of topics that the journal covers (in addition, of course, to his own speciality of climatology & atmospheric science). His deep experience in Wikipedia's inner workings and norms is also invaluable, given the WikiJournals' interactions with both the site and its community. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:17, 24 May 2023 (UTC) == Lint error on drafts == Lint errors never bother me. Moreover, I don't know that they bother anybody on Wikiversity. But I got into the habit of checking for lint errors and found one that appears any time a new preprint draft is created. See [[WikiJournal Preprints/Test page]]. This is strictly [[w:FYI|FYI]]. I am not asking or requesting anybody to fix this problem. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 07:36, 26 December 2023 (UTC) :I don't see this error. Can you point out which section showed the lint error? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 05:25, 29 December 2023 (UTC) == Using the new extension for inline comments == What about using this for peer review? https://www.mediawiki.org/wiki/User:JayanthVikash/GSoC/InlineComments-Draft Moreover, you may be interested in [[:c:Category:Public peer review]] such as [https://github.com/danielBingham/peerreview this open source repo] (demo website may be up at another time). I wondered why one can't provide peer review feedback on preprints on the major preprint publishing websites like arxiv – have you looked into integrating with these so that for example one could provide feedback here on these preprints rather than making this only about publishing of papers here? [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 17:59, 20 August 2024 (UTC) == One more preprint == Dear editors, I have finished my article "[[WikiJournal Preprints/Is there a relationship between volcanoes and earthquakes based on Wikidata?|Is there a relationship between volcanoes and earthquakes based on Wikidata?]]" and submitted the authorship declaration form. Should I now wait for a reviewer to be assigned, or is there any further action I need to take? -- [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 11:03, 10 February 2025 (UTC) :I don't know how to review it. But if you tell me how, I could step up (if I know enough about the matter which, without reading the article I can't tell) [[User:Jcintasr|Jcintasr]] ([[User talk:Jcintasr|discuss]] • [[Special:Contributions/Jcintasr|contribs]]) 06:03, 17 October 2025 (UTC) ::@[[User:Jcintasr|Jcintasr]] I am unable to comment on the paper because the page appears to be requested for deletion by the author {{u|AKA MBG}}. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:50, 2 January 2026 (UTC) :::Thank you for your concern, but I already published this article in another scientific journal seven months ago. [[User:AKA MBG|Andrew Krizhanovsky]] ([[User talk:AKA MBG|discuss]] • [[Special:Contributions/AKA MBG|contribs]]) 07:48, 3 January 2026 (UTC) ::::No problem. Thanks for getting back to us. Just out of curiosity, do you have the link to your published article? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:29, 4 January 2026 (UTC) == Formatting of articles == I noticed that the article format seems to have switched recently from two columns to only one. The previous two column format looks dramatically cleaner and more professional ([https://search.informit.org/doi/pdf/10.3316/informit.T2024050300027300774129743 like here]). Is there an option for your submitted paper to retain the two column format? Thanks ~ [[User:HAL333|HAL333]] ([[User talk:HAL333|discuss]] • [[Special:Contributions/HAL333|contribs]]) 22:11, 3 May 2025 (UTC) == Is Wikijournal of Science still active? == I've been thinking on update a couple articles in Wikipedia through the WikiJournal of Science's workflow and, if that works, maybe send an original research. I love the idea of this extremely open for all approach and I think that's what current science needs. Nonetheless, the project seems a bit dead without any new publication since January 2024.  Is it still active? [[User:Jcintasr|Jcintasr]] ([[User talk:Jcintasr|discuss]] • [[Special:Contributions/Jcintasr|contribs]]) 05:55, 17 October 2025 (UTC) :Same question here. I just submitted an article ([[w:Pentagram map]]), took me a lot of time, hoping it would be put on Wikiversity... [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 13:13, 8 December 2025 (UTC) ::How is the process going? Did you receive any feedback? [[User:Jcintasr|Jcintasr]] ([[User talk:Jcintasr|discuss]] • [[Special:Contributions/Jcintasr|contribs]]) 09:53, 21 December 2025 (UTC) :::Hello @[[User:Jcintasr|Jcintasr]], :::Unfortunately no, I didn't get any answer from the form that I filled, nor from the email I sent them. And from what @[[User:Francesco Cattafi|Francesco Cattafi]] told me [[Talk:WikiJournal User Group#Wikipedia:WikiJournal article nominations is dead|here]], he didn't got any answer in the past months either. Let's hope that the year 2026 (the 25th birthday of Wiki!) will bring so much energy that the WikiJournal will get back on track. :::Best, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 17:53, 22 December 2025 (UTC) ::::@[[User:Jcintasr|Jcintasr]] and @[[User:Regliste|Regliste]], sorry it's been a busy year on my real work side. We published [[WikiJournal of Science/Volume 7 Issue 1|8 papers in 2024]] and [[WikiJournal of Science/Volume 8 Issue 1|1 paper in 2025]]. I'm trying to work through the backlog for the past ~6 months. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:46, 2 January 2026 (UTC) :::::Hello @[[User:OhanaUnited|OhanaUnited]], and thanks a lot for your answer. Good luck with everything, and do not hesitate to communicate if the WikiJournal can be helped. Happy new year, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 13:19, 4 January 2026 (UTC) ::::::@[[User:Regliste|Regliste]] Thanks for the offer. Judging from your ORCID, can you confirm if your area of expertise is in mathematics? Do you have in reviewing this paper [[WikiJournal Preprints/Kinematics of the cuboctahedron]], or know colleagues who might be able to? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:09, 4 January 2026 (UTC) :::::::Hello @[[User:OhanaUnited|OhanaUnited]]. Yes, I confirm that my area of expertise is mathematics. I'm a PhD student, this is why my ORCID is quite empty for now. I don't know if with this, I'm qualified for reviewing [[WikiJournal Preprints/Kinematics of the cuboctahedron]]. However, I can write to some colleagues that might be up for it. What information should I send them, apart from [[WikiJournal of Science/Peer reviewers]] ? [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 11:22, 6 January 2026 (UTC) ::::::::It depends if you think you are qualified to give a review on Kinematics of the cuboctahedron, based on the instructions on [[WikiJournal of Science/Peer reviewers]]. If you do, please let me know. If not, no worries. We have message templates on [[WikiJournal of Science/Editorial guidelines/Message templates#Emails to peer reviewers]] for you to use to write to your colleagues. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:48, 9 January 2026 (UTC) :::::::::@[[User:Jcintasr|Jcintasr]] : A little message to confirm you that yes, the Wikijournal of Science is still alive. There have been two published articles in 2026 and more might come. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 15:24, 16 June 2026 (UTC) toedzng9doupf9lqiuxhwzrffda4kuv 0 207402 2816056 2815031 2026-06-17T01:27:28Z Jessephu 3079828 2816056 wikitext text/x-wiki =A= '''Alignment''' – The arrangement of elements in a design so they line up neatly. '''Aspect Ratio''' – The proportional relationship between an image's width and height (e.g., 16:9). =B= '''Bleed''' - The area outside the trim edge that ensures no unprinted edges appear. '''Brand Identity''' – Visual elements such as logos, colors, and typography that represent a brand. =C= '''CMYK''' - Cyan, Magenta, Yellow, Key (Black). CMYK is the standard colour mode used for printing. '''Cropping''' – Removing unwanted parts of an image. =D= '''DPI''' - Dots per Inch on a printed page. =E= =F= =G= '''Grid''' – A framework of lines used to organize content in a design. =H= '''Hierarchy''' - The arrangement of elements in a way that signifies importance. =I= =J= =K= '''Kerning -''' The spacing between individual characters in a word. =L= =M= '''Mockup -''' A realistic model of a design used for presentation or feedback. =N= '''Negative Space -''' The empty space around and between elements in a design. =O= =P= '''PPI''' - Pixels per Inch; number of pixels per inch in an image. '''PNG''' – An image format that supports transparent backgrounds. =Q= =R= '''Raster Image''' - an image made up of thousands of pixels. Also known as bitmap image. Photos are an example of a raster image. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> '''RGB''' - Red, Green, Blue. RGB is the colour mode used for screen output. =S= =T= '''Typography -''' The art of arranging text to make it legible and visually appealing. '''Tracking''' – Adjusting the spacing across a group of letters. =U= =V= '''Vector Image''' - an image made up of points, as opposed to raster images which are made from pixels. Each point has a defined X and Y coordinates. Vector images can be resized without loss of quality. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> =W= '''Wireframe -''' A basic layout of a design that outlines structure without detailed visuals. =X= =Y= =Z= =See Also= [http://www.superdream.co.uk/glossary-of-design-terms/ Superdream Glossary of Design Terms] =References= <div class="references-small"> <references/> </div> [[Category:Graphic design]] l49uq8rar2ex2gudlqkqjqcrmd0i2r0 0 213507 2815994 2799490 2026-06-16T18:31:33Z Watchduck 137431 /* Subgroups of order 8 */ 2815994 wikitext text/x-wiki [[File:Polyhedron pair 6-8.png|thumb|A {{w|compound of cube and octahedron}} with full octahedral symmetry]] [[File:Full octahedral group; cycle graph.svg|thumb|[[w:Cycle graph (algebra)|Cycle graph]] of O_h]] [[File:Symmetric group 4; Cayley table (left); numbers.svg|thumb|Cayley table of the [[Symmetric group S4|symmetric group S₄]]]] [[File:Symmetric group 3; Cayley table; matrices (left).svg|thumb|Cayley table of the symmetric group S₃ with 3×3 permutation matrices]] The '''[[w:Octahedral symmetry|full octahedral group]] O_h''' is the {{w2|hyperoctahedral group}} of dimension 3. This article mainly looks at it as the symmetry group of the cube. There are {{oeis|A000165}}(3) = 48 permutation of the cube. Half of them are its rotations, forming the subgroup O (the [[symmetric group S4|symmetric group S₄]]), and the other half are their inversions.<br> The [[w:Point reflection|inversion]] is the permutation that exchanges opposite vertices of the cube. <small>It is not to be confused with the [[Inversion (discrete mathematics)|inversion of a permutation]].</small> {| class=wikitable |+ {{w2|Conjugacy class}}es !colspan="3" style="border-right: 3px solid black;"|Elements of O !colspan="3"|Inversions of elements of O |- |identity |style="background:#fff;{{Text default color}};"|'''neut''' |style="background:#fff;{{Text default color}}; border-right: 3px solid black;"|'''0''' | inversion |style="background:#ffff7f;{{Text default color}};"|'''inv3''' |style="background:#ffff7f;{{Text default color}};"|'''0<nowiki>'</nowiki>''' |- |3 &times; rotation by 180° about a 4-fold axis |style="background:#fff;{{Text default color}};"|'''inv2''' |style="background:#fff;{{Text default color}}; border-right: 3px solid black;"|'''7, 16, 23''' |3 &times; reflection in a plane perpendicular to a 4-fold axis |style="background:#ffff7f;{{Text default color}};"|'''ref1''' |style="background:#ffff7f;{{Text default color}};"|'''7', 16', 23<nowiki>'</nowiki>''' |- |8 &times; rotation by 120° about a 3-fold axis |style="background:#fff;{{Text default color}};"|rot3 |style="background:#fff; {{Text default color}}; border-right: 3px solid black;"|3, 4, 8, 11, 12, 15, 19, 20 |8 &times; rotoreflection by 60° |style="background:#ffff7f;{{Text default color}};"|rotref3 |style="background:#ffff7f;{{Text default color}};"|3', 4', 8', 11', 12', 15', 19', 20' |- |6 &times; rotation by 180° about a 2-fold axis |style="background:#3375ff;{{Text default color}};"|rot2 |style="background:#3375ff;{{Text default color}}; border-right: 3px solid black;"|1', 2', 5', 6', 14', 21' |6 &times; reflection in a plane perpendicular to a 2-fold axis |style="background:#33d42a;{{Text default color}};"|ref2 |style="background:#33d42a;{{Text default color}};"|1, 2, 5, 6, 14, 21 |- |6 &times; rotation by 90° about a 4-fold axis |style="background:#ef2500;{{Text default color}};"|rot1 |style="background:#ef2500;{{Text default color}}; border-right: 3px solid black;"|9', 10', 13', 17', 18', 22' |6 &times; rotoreflection by 90° |style="background:#ffa200;{{Text default color}};"|rotref1 |style="background:#ffa200;{{Text default color}};"|9, 10, 13, 17, 18, 22 |} {| class="wikitable collapsible collapsed" style="text-align: center;" !colspan="5"| Examples |- | [[File:Cube permutation 0 0.svg|180px]]<br><math>(0,0) = 0</math> | [[File:Cube permutation 3 0.svg|180px]]<br><math>(3,0) = 7</math> | [[File:Cube permutation 0 3.svg|180px]]<br><math>(0,3) = 3</math> | [[File:Cube permutation 7 1.svg|180px]]<br><math>(7,1) = 1'</math> | [[File:Cube permutation 4 5.svg|180px]]<br><math>(4,5) = 9'</math> |- |style="background:#fff;{{Text default color}};"|'''neut''' |style="background:#fff;{{Text default color}};"|'''inv2''' |style="background:#fff;{{Text default color}};"|rot3 |style="background:#3375ff;{{Text default color}};"|rot2 |style="background:#ef2500;{{Text default color}};"|rot1 |- style="border-top: 3px solid black" |style="background:#ffff7f;{{Text default color}};"|'''inv3''' |style="background:#ffff7f;{{Text default color}};"|'''ref1''' |style="background:#ffff7f;{{Text default color}};"|rotref3 |style="background:#33d42a;{{Text default color}};"|ref2 |style="background:#ffa200;{{Text default color}};"|rotref1 |- | <math>(7,0) = 0'</math><br>[[File:Cube permutation 7 0.svg|180px]] | <math>(4,0) = 7'</math><br>[[File:Cube permutation 4 0.svg|180px]] | <math>(7,3) = 3'</math><br>[[File:Cube permutation 7 3.svg|180px]] | <math>(0,1) = 1</math><br>[[File:Cube permutation 0 1.svg|180px]] | <math>(3,5) = 9</math><br>[[File:Cube permutation 3 5.svg|180px]] |- |colspan="5"| See full list below: [[#8×6 matrix|8×6 matrix]] or [[#Conjugacy classes|Conjugacy classes]] |} As the [[w:hyperoctahedral group|hyperoctahedral group]] of dimension 3 the full octahedral group is the {{w2|wreath product}} <math>S_2 \wr S_3 \simeq S_2^3 \rtimes S_3</math>,<br>and a natural way to identify its elements is as pairs <math>(m, n)</math> with <math>m \in [0, 2^3)</math> and <math>n \in [0, 3!)</math>.<br>But as it is also the [[w:Direct product of groups|direct product]] <math>S_4 \times S_2</math>, one can simply identify the elements of tetrahedral subgrup ''T_d'' as <math>a \in [0, 4!)</math> and their inversions as <math>a'</math>. So e.g. the identity <math>(0, 0)</math> is represented as <math>0</math> and the inversion <math>(7, 0)</math> as <math>0'</math>.<br> <math>(3, 1)</math> is represented as <math>6</math> and <math>(4, 1)</math> as <math>6'</math>. A [[w:Improper rotation|rotoreflection]] is a combination of rotation and reflection.<br> <small>While a rotation leaves its axis and a reflection leaves its plane unchanged, a rotoreflection leaves only the center unchanged.</small> {| class="wikitable collapsible collapsed" style="text-align: center;" !colspan="5"| Illustration of rotoreflections |- | {{multiple image | align = left | image1 = Cube permutation 4 0.svg | width1 = 230 | caption1 = The reflection <math>7'</math> | image2 = Cube permutation 0 4.svg | width2 = 230 | caption2 = applied on the 120° rotation <math>4</math> | image3 = Cube permutation 4 4.svg | width3 = 230 | caption3 = gives the 60° rotoreflection <math>8'</math>. }} <br><math>7' \circ 4 = 8'</math> |- | {{multiple image | align = left | image1 = Cube permutation 4 0.svg | width1 = 230 | caption1 = The reflection <math>7'</math> | image2 = Cube permutation 1 1.svg | width2 = 230 | caption2 = applied on the 90° rotation <math>22'</math> | image3 = Cube permutation 5 1.svg | width3 = 230 | caption3 = gives the 90° rotoreflection <math>17</math>. }} <br><math>7' \circ 22' = 17</math> |} ==Cayley table== As there are two ways to denote the elements of this group, there are two ways to write the Cayley table. ===S₄=== The easier one is that in [[symmetric group S4|S₄]] based notation: It simply doubles the Cayley table of S₄.<br> With <math>a, b, c \in S_4</math> and <math>a', b', c'</math> being their respective inversions, <math>a \circ b = c</math> implies <math>a' \circ b' = c</math> and <math>a \circ b' = a' \circ b = c'</math>. Example: <math>12' \circ 21 = 18'</math> ===pairs=== The notation with pairs is probably more useful, but also more complicated: : <math>(pm, pn) = (am, an) \circ (bm, bn)</math> The calculation of <math>pm</math> involves the cube vertex permutation shown in the images. &nbsp; E.g. <math>\operatorname{cvp}(2, 3) = (2, 6, 3, 7, ~ 0, 4, 1, 5)</math>. : <math>pm = \operatorname{cvp}(am, an)[bm]</math> <math>pn</math> simply follows from the Cayley table of S₃: : <math>pn = an \circ bn</math> Example: <math>(pm, pn) = (2, 3) \circ (6, 2)</math> : <math>pm = \operatorname{cvp}(2, 3)[6] = 1</math> : <math>pn = 3 \circ 2 = 5</math><br> {| class="wikitable collapsible collapsed" style="text-align: center;" !example |- | {| | [[File:Cube permutation 2 3.svg|230px|border]] | | [[File:Cube permutation 6 2.svg|230px|border]] | | [[File:Cube permutation 1 5.svg|230px|border]] |- | <math>12'</math> |rowspan="2"| <math>\circ</math> | <math>21</math> |rowspan="2"| <math>=</math> | <math>18'</math> |- | <math>(2, 3)</math> | <math>(6, 2)</math> | <math>(1, 5)</math> |} |} ==Overview== ===Truncated cuboctahedron=== The vertices of the {{w|truncated cuboctahedron}} correspond to the elements of this group. Each of its faces of its dual, the {{w|disdyakis dodecahedron}}, is a {{w|fundamental domain}}. {| style="width: 100%;" | [[File:Full octahedral group elements in truncated cuboctahedron; numbers.svg|280px]] | [[File:Full octahedral group elements in truncated cuboctahedron; JF.png|320px]] | [[File:Polyhedron great rhombi 6-8 max.png|280px]] | [[File:Polyhedron great rhombi 6-8 dual max.png|280px]] |} ===8×6 matrix=== {| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #ddd;" ! bgcolor="#eee"| |- | {{Full octahedral group; image matrices}} <!--START NESTED BOX 1--> {| class="collapsible collapsed" style="width: 100%; border: 1px solid #999; border-bottom:none;" !bgcolor="white"| truncated cuboctahedron |- |{{Full octahedral group; image matrices; truncated octahedron}} |}<!--END NESTED BOX 1--> <!--START NESTED BOX 2--> {| class="collapsible collapsed" style="width: 100%; border: 1px solid #999; background-color: white;" !bgcolor="white"| number matrices |- |{{Full octahedral group; text matrices}} |- |style="text-align:left;"| It can be seen that the reverse colexicographic ranks of two inversions <small>(in the sense of [[w:Point reflection|point reflections]])</small> add up to 40319 - the rank of ''the inversion'' <small>(in the sense of point reflection)</small> itself.<br> This means that their [[Inversion (discrete mathematics)|inversion sets]] <small>(sets of pairs out of their natural order)</small> are complements. (Beware the two different meanings of ''inversion'' here.) |}<!--END NESTED BOX 2--> |} ===Hexagon corresponding to top matrix row=== {| class="collapsible open" style="width: 100%; border: 1px solid #ddd; border-bottom:0;" ! bgcolor="#eee"| 3D diagrams |- | The files below illustrate the subgroup '''C_3v''' or '''[3]''' that corresponds to the top matrix row. It contains the six permutations of the cube that leave the main diagonal fixed. {| style="width:100%; text-align:center;" | [[File:Permutohedron order 3 in cube, 1-based.png|thumb|center|200px|permutohedron coordinates]] |rowspan="2"| [[File:Permutohedron of S3; cube permutations.png|thumb|center|400px|[[w:permutohedron|permutohedron]] ]] |rowspan="2"| [[File:Cayley graph of S3 with cube permutations; generators a, b.png|thumb|center|400px|[[w:Cayley graph|Cayley graph]] generated by [[File:Finite permutation number 1.svg|20px]] and [[File:Finite permutation number 2.svg|20px]]]] |rowspan="2"| [[File:Cayley graph of S3 with cube permutations; generators a, r.png|thumb|center|400px|Cayley graph generated by [[File:Finite permutation number 1.svg|20px]] and [[File:Finite permutation number 4.svg|20px]]]] |- | [[File:Subgroup of Oh; S3 green 03; example solid.png|thumb|center|150px|example solid]] |} |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom:0;" ! bgcolor="#eee"| 2D equivalents |- | The rest of this article uses left [[w:group action|action]], i.e. <math>f \cdot g</math> means ''first <math>g</math>, then <math>f</math>'',<br> but these two-dimensional diagrams use right action, i.e. <math>f \cdot g</math> means ''first <math>f</math>, then <math>g</math>''.<br> So <math>{\color{RoyalBlue}a} ~ \widehat{=}</math> [[File:Finite permutation number 1.svg|20px]] and <math>{\color{RubineRed}r} ~ \widehat{=}</math> [[File:Finite permutation number 4.svg|20px]], but <math>{\color{RoyalBlue}a} \cdot {\color{RubineRed}r} ~ \widehat{=}</math> [[File:Finite permutation number 4.svg|20px]]<math>\cdot</math>[[File:Finite permutation number 1.svg|20px]]<math>=</math>[[File:Finite permutation number 5.svg|20px]] and <math>{\color{RubineRed}r} \cdot {\color{RoyalBlue}a} ~ \widehat{=}</math> [[File:Finite permutation number 1.svg|20px]]<math>\cdot</math>[[File:Finite permutation number 4.svg|20px]]<math>=</math>[[File:Finite permutation number 2.svg|20px]]. {| style="width:100%; text-align:center;" | [[File:Permutohedron of S3 with permutation results as coordinates; numbers.svg|thumb|center|200px|permutohedron]] | [[File:Permutohedron of S3 with permutation results as coordinates; triangles.svg|thumb|center|400px|permutohedron]] | [[File:Cayley graph of S3 with triangles; generators a, b.svg|thumb|center|400px|Cayley graph generated by two reflections]] | [[File:Cayley graph of S3 with triangles; generators a, r.svg|thumb|center|400px|Cayley graph generated by reflection and rotation]] |} |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd;" !colspan="2" bgcolor="#eee"| details |- |style="width:260px;"| [[File:Subgroup of Oh; S3 green 03; example solid (triangle).png|thumb|center|200px|permuted triangle in the cube (alternative example solid)]] | {{Full octahedral group; hexagon}} |- |colspan="2"| This is left action again, so the 3×3 permutation matrices shown here are the [[w:Transpose|transposes]] of those in the small permutohedron in the box above - which only makes a difference for [[File:Finite permutation number 3.svg|20px]] and [[File:Finite permutation number 4.svg|20px]]. |} ===Cubes corresponding to matrix columns=== Each of the six cubes in the following collapsible boxes shows one of the basic permutations from the top row of the matrix in the bottom left position.<br> In the other seven positions are the products of applying the reflections along coordinate axes on these basic permutations. {{Full octahedral group; cube of cube perms|0}} {{Full octahedral group; cube of cube perms|1}} {{Full octahedral group; cube of cube perms|2}} {{Full octahedral group; cube of cube perms|3}} {{Full octahedral group; cube of cube perms|4}} {{Full octahedral group; cube of cube perms|5}} ===Conjugacy classes=== The full octahedral group has {{oeis|A000712}}(3) = 10 conjugacy classes. Two permutations <math>(m, n)</math> and <math>(k, n)</math> are complementary to each other, if <math>m + k = 7</math>.<br> Complementary permutations sum up to a vector of 7s, and their [[inversion (discrete mathematics)|inversion sets]] are {{w|Complement (set theory)|complements}},<br> so their inversion numbers sum up to 28. <small>(Compare one of the number matrices [[#8×6 matrix|above]].)</small> The conjugacy classes below are always shown in complementary pairs (like inv2/ref1 or rot2/ref2).<br> The numbers over the triangles are the inversion numbers of the corresponding permutations. It can be seen that corresponding numbers add up to 28. {| class="wikitable" style="width: 100%; text-align: center;" |- !style="background: #ffffff;"| neut <small>(1)</small> !style="background: #ffff7f; border-right: 3px solid #a2a9b1;"| inv3 <small>(1)</small> !style="background: #ffffff;"| inv2 <small>(3)</small> !style="background: #ffff7f; border-right: 3px solid #a2a9b1;"| ref1 <small>(3)</small> !style="background: #ffffff;"| rot3 <small>(8)</small> !style="background: #ffff7f; border-right: 3px solid #a2a9b1;"| rotref3 <small>(8)</small> !style="background: #3375ff;"| rot2 <small>(6)</small> !style="background: #33d42a; border-right: 3px solid #a2a9b1;"| ref2 <small>(6)</small> !style="background: #ef2500;"| rot1 <small>(6)</small> !style="background: #ffa200;"| rotref1 <small>(6)</small> |- | [[File:Full octahedral group; set partition neut.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition inv3.svg|120px]] | [[File:Full octahedral group; set partition inv2 0.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition ref1 0.svg|120px]] | [[File:Full octahedral group; set partition rot3 0.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition rotref3 0.svg|120px]] | [[File:Full octahedral group; set partition rot2 0b.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Full octahedral group; set partition ref2 0b.svg|120px]] | [[File:Full octahedral group; set partition rot1 0.svg|120px]] | [[File:Full octahedral group; set partition rotref1 0.svg|120px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !style="width: 50%; background: #fff; border-right: 3px solid black;"| neut <small>(1 × 1)</small> !style="width: 50%; background: #ffff7f;"| inv3 <small>(1 × 1)</small> |- |style="border-right: 3px solid black;"| 0<br>[[File:Full octahedral group; set partition neut.svg|180px]] | 28<br>[[File:Full octahedral group; set partition inv3.svg|180px]] |- |style="border-right: 3px solid black;"| [[File:Cube permutation 0 0.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 7 0.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="3" style="width: 50%; background: #fff; border-right: 3px solid black;"| inv2 <small>(3 × 1)</small> !colspan="3" style="width: 50%; background: #ffff7f;"| ref1 <small>(3 × 1)</small> |- | 12<br>[[File:Full octahedral group; set partition inv2 0.svg|180px]] | 20<br>[[File:Full octahedral group; set partition inv2 1.svg|180px]] |style="border-right: 3px solid black;"| 24<br>[[File:Full octahedral group; set partition inv2 2.svg|180px]] | 16<br>[[File:Full octahedral group; set partition ref1 2.svg|180px]] | 8<br>[[File:Full octahedral group; set partition ref1 1.svg|180px]] | 4<br>[[File:Full octahedral group; set partition ref1 0.svg|180px]] |- | [[File:Cube permutation 3 0.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 5 0.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |style="border-right: 3px solid black;"| [[File:Cube permutation 6 0.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 4 0.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 2 0.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 1 0.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="8" style="width: 50%; background: #fff;"| rot3 <small>(4 × 2)</small> |- |colspan="2" style="border-right: 3px solid #a2a9b1;"| 6<br>[[File:Full octahedral group; set partition rot3 0.svg|180px]] |colspan="2" style="border-right: 3px solid #a2a9b1;"| 14<br>[[File:Full octahedral group; set partition rot3 1.svg|180px]] |colspan="2" style="border-right: 3px solid #a2a9b1;"| 18<br>[[File:Full octahedral group; set partition rot3 2.svg|180px]] |colspan="2"| 18<br>[[File:Full octahedral group; set partition rot3 3.svg|180px]] |- | [[File:Cube permutation 0 3.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 3.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 0 4.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 5 3.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 3.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 3 4.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 3 3.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 3.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 6 4.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 6 3.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 3.svg|20px]] | [[File:Cube permutation 5 4.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 4.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="8" style="width: 50%; background: #ffff7f;"| rotref3 <small>(4 × 2)</small> |- |colspan="2" style="border-right: 3px solid #a2a9b1;"| 22<br>[[File:Full octahedral group; set partition rotref3 0.svg|180px]] |colspan="2" style="border-right: 3px solid #a2a9b1;"| 14<br>[[File:Full octahedral group; set partition rotref3 1.svg|180px]] |colspan="2" style="border-right: 3px solid #a2a9b1;"| 10<br>[[File:Full octahedral group; set partition rotref3 2.svg|180px]] |colspan="2"| 10<br>[[File:Full octahedral group; set partition rotref3 3.svg|180px]] |- | [[File:Cube permutation 7 3.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 3.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 7 4.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 2 3.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 3.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 4 4.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 4 3.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 3.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 1 4.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 1 3.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 3.svg|20px]] | [[File:Cube permutation 2 4.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 4.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="6" style="width: 50%; background: #3375ff; border-right: 3px solid black;"| rot2 <small>(6 × 1)</small> |- | 26<br>[[File:Full octahedral group; set partition rot2 2b.svg|180px]] |style="border-right: 3px solid #a2a9b1;"| 18<br>[[File:Full octahedral group; set partition rot2 2a.svg|180px]] | 20<br>[[File:Full octahedral group; set partition rot2 1b.svg|180px]] |style="border-right: 3px solid #a2a9b1;"| 12<br>[[File:Full octahedral group; set partition rot2 1a.svg|180px]] | 24<br>[[File:Full octahedral group; set partition rot2 0b.svg|180px]] | 8<br>[[File:Full octahedral group; set partition rot2 0a.svg|180px]] |- | [[File:Cube permutation 7 1.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 4 1.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 7 5.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 5.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 2 5.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 5.svg|20px]] | [[File:Cube permutation 7 2.svg|100px|border]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 2.svg|20px]] | [[File:Cube permutation 1 2.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 2.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="6" style="width: 50%; background: #33d42a;"| ref2 <small>(6 × 1)</small> |- | 2<br>[[File:Full octahedral group; set partition ref2 0a.svg|180px]] |style="border-right: 3px solid #a2a9b1;"| 10<br>[[File:Full octahedral group; set partition ref2 0b.svg|180px]] | 8<br>[[File:Full octahedral group; set partition ref2 1a.svg|180px]] |style="border-right: 3px solid #a2a9b1;"| 16<br>[[File:Full octahedral group; set partition ref2 1b.svg|180px]] | 4<br>[[File:Full octahedral group; set partition ref2 2a.svg|180px]] | 20<br>[[File:Full octahedral group; set partition ref2 2b.svg|180px]] |- | [[File:Cube permutation 0 1.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 3 1.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 0 5.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 5.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Cube permutation 5 5.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 5.svg|20px]] | [[File:Cube permutation 0 2.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 2.svg|20px]] | [[File:Cube permutation 6 2.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 2.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="6" style="width: 50%; background: #ef2500; border-right: 3px solid black;"| rot1 <small>(3 × 2)</small> !colspan="6" style="width: 50%; background: #ffa200;"| rotref1 <small>(3 × 2)</small> |- |colspan="2"| 6<br>[[File:Full octahedral group; set partition rot1 0.svg|180px]] |colspan="2"| 12<br>[[File:Full octahedral group; set partition rot1 1.svg|180px]] |colspan="2" style="border-right: 3px solid black;"| 12<br>[[File:Full octahedral group; set partition rot1 2.svg|180px]] |colspan="2"| 22<br>[[File:Full octahedral group; set partition rotref1 0.svg|180px]] |colspan="2"| 16<br>[[File:Full octahedral group; set partition rotref1 1.svg|180px]] |colspan="2"| 16<br>[[File:Full octahedral group; set partition rotref1 2.svg|180px]] |- | [[File:Cube permutation 2 1.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 1 1.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 1 5.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 5.svg|20px]] | [[File:Cube permutation 4 5.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 5.svg|20px]] | [[File:Cube permutation 4 2.svg|100px|border]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 2.svg|20px]] |style="border-right: 3px solid black;"| [[File:Cube permutation 2 2.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 2.svg|20px]] | [[File:Cube permutation 5 1.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 6 1.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 6 5.svg|100px|border]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 5.svg|20px]] | [[File:Cube permutation 3 5.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 5.svg|20px]] | [[File:Cube permutation 3 2.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 2.svg|20px]] | [[File:Cube permutation 5 2.svg|100px|border]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 2.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%; text-align: center;" |- !colspan="8"| Conjugacy classes of square permutations |- !style="border-right: 3px solid #a2a9b1;"| neut !style="border-right: 3px solid #a2a9b1;"| inv2 !style="border-right: 3px solid #a2a9b1;" colspan="2"| ref1 !style="border-right: 3px solid #a2a9b1;" colspan="2"| ref2 !colspan="2"| rot1 |- |style="border-right: 3px solid #a2a9b1;"| 0<br>[[File:Dihedral group Dih 4; set partition neut.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| 6<br>[[File:Dihedral group Dih 4; set partition inv2.svg|120px]] | 2<br>[[File:Dihedral group Dih 4; set partition ref1 0.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| 4<br>[[File:Dihedral group Dih 4; set partition ref1 1.svg|120px]] | 1<br>[[File:Dihedral group Dih 4; set partition ref2 a.svg|120px]] |style="border-right: 3px solid #a2a9b1;"| 5<br>[[File:Dihedral group Dih 4; set partition ref2 b.svg|120px]] |colspan="2"| 3<br>[[File:Dihedral group Dih 4; set partition rot1.svg|120px]] |- |style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 0 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 3 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Square permutation 1 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 2 0, colored.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Square permutation 0 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |style="border-right: 3px solid #a2a9b1;"| [[File:Square permutation 3 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Square permutation 1 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Square permutation 2 1, colored.svg|100px|border]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |} {| class="wikitable collapsible collapsed" style="width: 100%;" |- ! Examples for tesseract and penteract |- | The respective numbers of conjugacy classes for 4 and 5 dimensions are 20 and 36.<br> The following dictionaries show an example pair and the corresponding permutation from each conjugacy class.<br> (See [https://pastebin.com/VuKq29SV here] for the whole tesseract group.) <syntaxhighlight lang="python"> examples4 = { (0, 0): (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15), (0, 1): (0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15), (0, 3): (0, 1, 4, 5, 8, 9, 12, 13, 2, 3, 6, 7, 10, 11, 14, 15), (0, 7): (0, 2, 1, 3, 8, 10, 9, 11, 4, 6, 5, 7, 12, 14, 13, 15), (0, 9): (0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15), (1, 0): (1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14), (1, 1): (1, 3, 0, 2, 5, 7, 4, 6, 9, 11, 8, 10, 13, 15, 12, 14), (1, 2): (1, 0, 3, 2, 9, 8, 11, 10, 5, 4, 7, 6, 13, 12, 15, 14), (1, 3): (1, 3, 5, 7, 0, 2, 4, 6, 9, 11, 13, 15, 8, 10, 12, 14), (1, 7): (1, 3, 0, 2, 9, 11, 8, 10, 5, 7, 4, 6, 13, 15, 12, 14), (1, 8): (1, 0, 5, 4, 9, 8, 13, 12, 3, 2, 7, 6, 11, 10, 15, 14), (1, 9): (1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14), (3, 0): (3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12), (3, 2): (3, 2, 7, 6, 1, 0, 5, 4, 11, 10, 15, 14, 9, 8, 13, 12), (3, 6): (3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 15, 14, 13, 12), (3, 8): (3, 2, 7, 6, 11, 10, 15, 14, 1, 0, 5, 4, 9, 8, 13, 12), (3, 16): (3, 7, 11, 15, 2, 6, 10, 14, 1, 5, 9, 13, 0, 4, 8, 12), (7, 0): (7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8), (7, 6): (7, 6, 5, 4, 15, 14, 13, 12, 3, 2, 1, 0, 11, 10, 9, 8), (15, 0): (15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0) } examples5 = { (0, 0): (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31), (0, 1): (0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31), (0, 3): (0, 1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31), (0, 7): (0, 1, 4, 5, 2, 3, 6, 7, 16, 17, 20, 21, 18, 19, 22, 23, 8, 9, 12, 13, 10, 11, 14, 15, 24, 25, 28, 29, 26, 27, 30, 31), (0, 9): (0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31), (0, 27): (0, 2, 1, 3, 8, 10, 9, 11, 16, 18, 17, 19, 24, 26, 25, 27, 4, 6, 5, 7, 12, 14, 13, 15, 20, 22, 21, 23, 28, 30, 29, 31), (0, 33): (0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31), (1, 0): (1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30), (1, 1): (1, 3, 0, 2, 5, 7, 4, 6, 9, 11, 8, 10, 13, 15, 12, 14, 17, 19, 16, 18, 21, 23, 20, 22, 25, 27, 24, 26, 29, 31, 28, 30), (1, 2): (1, 0, 3, 2, 5, 4, 7, 6, 17, 16, 19, 18, 21, 20, 23, 22, 9, 8, 11, 10, 13, 12, 15, 14, 25, 24, 27, 26, 29, 28, 31, 30), (1, 3): (1, 3, 5, 7, 0, 2, 4, 6, 9, 11, 13, 15, 8, 10, 12, 14, 17, 19, 21, 23, 16, 18, 20, 22, 25, 27, 29, 31, 24, 26, 28, 30), (1, 7): (1, 3, 0, 2, 5, 7, 4, 6, 17, 19, 16, 18, 21, 23, 20, 22, 9, 11, 8, 10, 13, 15, 12, 14, 25, 27, 24, 26, 29, 31, 28, 30), (1, 8): (1, 0, 3, 2, 9, 8, 11, 10, 17, 16, 19, 18, 25, 24, 27, 26, 5, 4, 7, 6, 13, 12, 15, 14, 21, 20, 23, 22, 29, 28, 31, 30), (1, 9): (1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 27, 29, 31, 16, 18, 20, 22, 24, 26, 28, 30), (1, 26): (1, 0, 5, 4, 3, 2, 7, 6, 17, 16, 21, 20, 19, 18, 23, 22, 9, 8, 13, 12, 11, 10, 15, 14, 25, 24, 29, 28, 27, 26, 31, 30), (1, 27): (1, 3, 5, 7, 0, 2, 4, 6, 17, 19, 21, 23, 16, 18, 20, 22, 9, 11, 13, 15, 8, 10, 12, 14, 25, 27, 29, 31, 24, 26, 28, 30), (1, 31): (1, 3, 0, 2, 9, 11, 8, 10, 17, 19, 16, 18, 25, 27, 24, 26, 5, 7, 4, 6, 13, 15, 12, 14, 21, 23, 20, 22, 29, 31, 28, 30), (1, 32): (1, 0, 5, 4, 9, 8, 13, 12, 17, 16, 21, 20, 25, 24, 29, 28, 3, 2, 7, 6, 11, 10, 15, 14, 19, 18, 23, 22, 27, 26, 31, 30), (1, 33): (1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30), (3, 0): (3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12, 19, 18, 17, 16, 23, 22, 21, 20, 27, 26, 25, 24, 31, 30, 29, 28), (3, 2): (3, 2, 7, 6, 1, 0, 5, 4, 11, 10, 15, 14, 9, 8, 13, 12, 19, 18, 23, 22, 17, 16, 21, 20, 27, 26, 31, 30, 25, 24, 29, 28), (3, 6): (3, 2, 1, 0, 7, 6, 5, 4, 19, 18, 17, 16, 23, 22, 21, 20, 11, 10, 9, 8, 15, 14, 13, 12, 27, 26, 25, 24, 31, 30, 29, 28), (3, 8): (3, 2, 7, 6, 11, 10, 15, 14, 1, 0, 5, 4, 9, 8, 13, 12, 19, 18, 23, 22, 27, 26, 31, 30, 17, 16, 21, 20, 25, 24, 29, 28), (3, 16): (3, 7, 11, 15, 2, 6, 10, 14, 1, 5, 9, 13, 0, 4, 8, 12, 19, 23, 27, 31, 18, 22, 26, 30, 17, 21, 25, 29, 16, 20, 24, 28), (3, 26): (3, 2, 7, 6, 1, 0, 5, 4, 19, 18, 23, 22, 17, 16, 21, 20, 11, 10, 15, 14, 9, 8, 13, 12, 27, 26, 31, 30, 25, 24, 29, 28), (3, 30): (3, 2, 1, 0, 11, 10, 9, 8, 19, 18, 17, 16, 27, 26, 25, 24, 7, 6, 5, 4, 15, 14, 13, 12, 23, 22, 21, 20, 31, 30, 29, 28), (3, 32): (3, 2, 7, 6, 11, 10, 15, 14, 19, 18, 23, 22, 27, 26, 31, 30, 1, 0, 5, 4, 9, 8, 13, 12, 17, 16, 21, 20, 25, 24, 29, 28), (3, 40): (3, 7, 11, 15, 2, 6, 10, 14, 19, 23, 27, 31, 18, 22, 26, 30, 1, 5, 9, 13, 0, 4, 8, 12, 17, 21, 25, 29, 16, 20, 24, 28), (7, 0): (7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24), (7, 6): (7, 6, 5, 4, 15, 14, 13, 12, 3, 2, 1, 0, 11, 10, 9, 8, 23, 22, 21, 20, 31, 30, 29, 28, 19, 18, 17, 16, 27, 26, 25, 24), (7, 24): (7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 31, 30, 29, 28, 27, 26, 25, 24), (7, 30): (7, 6, 5, 4, 15, 14, 13, 12, 23, 22, 21, 20, 31, 30, 29, 28, 3, 2, 1, 0, 11, 10, 9, 8, 19, 18, 17, 16, 27, 26, 25, 24), (7, 60): (7, 6, 15, 14, 23, 22, 31, 30, 5, 4, 13, 12, 21, 20, 29, 28, 3, 2, 11, 10, 19, 18, 27, 26, 1, 0, 9, 8, 17, 16, 25, 24), (15, 0): (15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16), (15, 24): (15, 14, 13, 12, 11, 10, 9, 8, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16), (31, 0): (31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0) } </syntaxhighlight> |} ==Subgroups== O_h has 98 individual subgroups, which are all shown in the list below. <small>(A Python dictionary of them can be found [https://github.com/watchduck/full_octahedral_group/blob/master/projects/p03_subgroups/store_dicts.py here].)</small> They naturally divide in 33 bundles of similar subgroups, whose elements belong to the same [[w:conjugacy class|conjugacy classes]].<br> In this article these bundles are given naive names based on some of the colors assigned to their elements (like '''Dih₄ green orange''').<br> Each of them has a collapsible box below, containing representations of the individual subgroups. These belong to 25 bigger bundles, which can be identified with a label in [[w:Schoenflies notation|Schoenflies]] or [[w:Coxeter notation|Coxeter]] notation (like '''D_2d''' or '''[2⁺,4]''').<br> Each of them has a vertex in the big Hasse diagram below. Four different kinds of Coxeter notation can be distinguished, based on where they contain plus signs: {| class="wikitable" |- style="background: #f0e0f0;" | '''[...]⁺''' || rotate |- style="background: #e0f0f0;" | '''[...]''' || reflect |- style="background: #e0e0e0;" | '''[...⁺,...⁺]''' || cross |- style="background: #f0f0e0;" | '''[...⁺, ...]''' || mixed |} ===Hasse diagrams=== {| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #ddd; border-bottom: 0;" ! bgcolor="#eee"| All 25 bundles of similar subgroups |- | [[File:Full octahedral group; subgroups Hasse diagram.svg|center|1000px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #ddd; border-bottom: 0;" ! colspan="4" bgcolor="#eee"| tetrahedral, rotational and reflective subgroups |- | [[File:Symmetric group S4; lattice of subgroups Hasse diagram; 11 different cycle graphs.svg|thumb|center|300px|(Subgroups of S₄ as an abstract group)]] | [[File:Full octahedral group; subgroups Hasse diagram; Td.svg|thumb|center|300px|Subgroups of S₄ as the tetrahedral group '''T_d''' or '''[3,3]''']] | [[File:Full octahedral group; subgroups Hasse diagram; rotational.svg|thumb|center|300px|Rotational subgroups<br>Subgroups of S₄ as the chiral octahedral group '''O''' or '''[4,3]⁺''']] | [[File:Full octahedral group; subgroups Hasse diagram; reflective.svg|thumb|center|300px|Reflective subgroups<br>Coxeter notation is like in the file on the left without the plus signs. But there is no '''[2,3]''', and no green edge from '''[2,2]''' to '''[3,3]'''.]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #ddd;" ! colspan="2" bgcolor="#eee"| chiral tetrahedral and pyritohedral subgroups |- | [[File:Full octahedral group; subgroups Hasse diagram; T.svg|thumb|center|260px|Subgroups of [[w:Alternating group|A]]₄ as the chiral tetrahedral group '''T''' or '''[3,3]⁺''']] | [[File:Full octahedral group; subgroups Hasse diagram; Th.svg|thumb|center|550px|Subgroups of A₄×C₂ as the pyritohedral group '''T_h''' or '''[3⁺,4]''']] |} ===List of all subgroups=== :''For the same list including all permutations of the respective example solids, see [[Full octahedral group/List of all subgroups]].'' ====The group itself==== {| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #999;" !colspan="3" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">O_h</span> S₄ × C₂ <span style="float:right; padding-right:50px;">[4,3]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S4xC2; example solid.png|230px]] | [[File:Full octahedral group; cycle graph.svg|700px]] |rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 max.png|thumb|230px|{{w|truncated cuboctahedron}} ]] |- | [[File:Subgroup of Oh; S4xC2; matrix.svg|50px]] [[File:Subgroup of Oh; S4xC2; solid.png|80px]] |} ====Subgroups of order 24==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" !colspan="3" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">T_d</span> S₄ green orange <span style="float:right; padding-right:50px;">[3,3]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S4 green orange; example solid.png|230px]] | [[File:Subgroup of Oh; S4 green orange; cycle graph.svg|700px]] |rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid tetrahedral maxmatch.png|thumb|230px|nonuniform {{w|truncated octahedron}} ]] |- | [[File:Subgroup of Oh; S4 green orange; matrix.svg|50px|]] [[File:Subgroup of Oh; S4 green orange; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" !colspan="3" bgcolor="#f0e0f0"|<span style="float:left; padding-left:50px;">O</span> S₄ blue red <span style="float:right; padding-right:50px;">[4,3]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S4 blue red; example solid.png|230px]] | [[File:Subgroup of Oh; S4 blue red; cycle graph.svg|700px]] |rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png|thumb|230px|nonuniform {{w|snub cube}} ]] |- | [[File:Subgroup of Oh; S4 blue red; matrix.svg|50px|]] [[File:Subgroup of Oh; S4 blue red; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" !colspan="3" bgcolor="#f0f0e0"|<span style="float:left; padding-left:50px;">T_h</span> A₄ × C₂ <span style="float:right; padding-right:50px;">[3⁺,4]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; A4xC2; example solid.png|230px]] | [[File:Subgroup of Oh; A4xC2; cycle graph.svg|550px]] |rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid pyritohedral maxmatch.png|thumb|230px|cantic snub octahedron]] |- | [[File:Subgroup of Oh; A4xC2; matrix.svg|50px|]] [[File:Subgroup of Oh; A4xC2; solid.png|80px]] |} ====Subgroups of order 16==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">D_4h</span> Dih₄ × C₂ <span style="float:right; padding-right:50px;">[2,4]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4xC2 07; example solid.png|230px]] | [[File:Subgroup of Oh; Dih4xC2 07; cycle graph.svg|400px|]] | [[File:Subgroup of Oh; Dih4xC2 16; cycle graph.svg|400px|]] | [[File:Subgroup of Oh; Dih4xC2 23; cycle graph.svg|400px|]] |- | [[File:Subgroup of Oh; Dih4xC2 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4xC2 07; solid.png|80px]] | [[File:Subgroup of Oh; Dih4xC2 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4xC2 16; solid.png|80px]] | [[File:Subgroup of Oh; Dih4xC2 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4xC2 23; solid.png|80px]] |} ====Subgroups of order 12==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" !colspan="3" bgcolor="#f0e0f0"|<span style="float:left; padding-left:50px;">T</span> A₄ <span style="float:right; padding-right:50px;">[3,3]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; A4; example solid.png|230px]] | [[File:Subgroup of Oh; A4; cycle graph.svg|450px]] |rowspan="2" style="width:240px;"| [[File:Polyhedron great rhombi 6-8 subsolid 20 maxmatch.png|thumb|230px|nonuniform {{w|snub octahedron}} (i.e. {{w|Icosahedron#Pyritohedral_symmetry|icosahedron}}) ]] |- | [[File:Subgroup of Oh; A4; matrix.svg|50px|]] [[File:Subgroup of Oh; A4; solid.png|80px]] |- |colspan="3"|<!--START NESTED BOX--> {| class="collapsible collapsed" style="width: 100%; border: 1px solid #999;" !bgcolor="white"| submatrix |- |{{Full octahedral group; submatrix A4}} |}<!--END NESTED BOX--> |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0f0e0" colspan="5"|<span style="float:left; padding-left:50px;">D_3d</span> Dih₆ <span style="float:right; padding-right:50px;">[2⁺,6]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih6 03; example solid.png|230px]] | [[File:Subgroup of Oh; Dih6 03; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih6 11; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih6 15; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih6 08; cycle graph.svg|200px|]] |- | [[File:Subgroup of Oh; Dih6 03; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 03; solid.png|80px]] | [[File:Subgroup of Oh; Dih6 11; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 11; solid.png|80px]] | [[File:Subgroup of Oh; Dih6 15; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 15; solid.png|80px]] | [[File:Subgroup of Oh; Dih6 08; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih6 08; solid.png|80px]] |- |colspan="5"|<!--START NESTED BOX--> {| class="collapsible collapsed" style="width: 100%; border: 1px solid #999;" !bgcolor="white"| submatrix |- |{{Full octahedral group; submatrix Dih6 03}} |}<!--END NESTED BOX--> |} ====Subgroups of order 8==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" !colspan="2" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">D_2h</span> C₂³ white <span style="float:right; padding-right:50px;">[2,2]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2^3 white; example solid.png|230px]] | [[File:Subgroup of Oh; C2^3 white; cycle graph.svg|230px]] |- | [[File:Subgroup of Oh; C2^3 white; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 white; solid.png|80px|]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">D_2h</span> C₂³ green <span style="float:right; padding-right:50px;">[2,2]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2^3 green 07; example solid.png|230px]] | [[File:Subgroup of Oh; C2^3 green 07; cycle graph.svg|220px|]] | [[File:Subgroup of Oh; C2^3 green 16; cycle graph.svg|220px|]] | [[File:Subgroup of Oh; C2^3 green 23; cycle graph.svg|220px|]] |- | [[File:Subgroup of Oh; C2^3 green 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 green 07; solid.png|80px|]] | [[File:Subgroup of Oh; C2^3 green 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 green 16; solid.png|80px|]] | [[File:Subgroup of Oh; C2^3 green 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C2^3 green 23; solid.png|80px|]] |} <small>([[#Symmetry group of the cuboid|Below]] the C₂³ subgroups are shown in more detail.)</small> {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">C_4h</span> C₄ × C₂ <span style="float:right; padding-right:50px;">[4⁺,2]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C4xC2 07; example solid.png|230px]] | [[File:Subgroup of Oh; C4xC2 07; cycle graph.svg|350px|]] | [[File:Subgroup of Oh; C4xC2 16; cycle graph.svg|350px|]] | [[File:Subgroup of Oh; C4xC2 23; cycle graph.svg|350px|]] |- | [[File:Subgroup of Oh; C4xC2 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C4xC2 07; solid.png|80px|]] | [[File:Subgroup of Oh; C4xC2 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C4xC2 16; solid.png|80px|]] | [[File:Subgroup of Oh; C4xC2 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C4xC2 23; solid.png|80px|]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C_4v</span> Dih₄ green red <span style="float:right; padding-right:50px;">[4]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 green red 07; example solid.png|230px]] | [[File:Subgroup of Oh; Dih4 green red 07; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 green red 16; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 green red 23; cycle graph.svg|200px|]] |- | [[File:Subgroup of Oh; Dih4 green red 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green red 07; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 green red 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green red 16; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 green red 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green red 23; solid.png|80px|]] |- |colspan="4"|<!--START NESTED BOX--> {| class="collapsible collapsed" style="width: 100%; border: 1px solid #999;" !bgcolor="white"| submatrix |- |{{Full octahedral group; submatrix Dih4 green red 07}} |}<!--END NESTED BOX--> |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">D_2d</span> Dih₄ blue orange <span style="float:right; padding-right:50px;">[2⁺,4]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 blue orange 07; example solid.png|230px]] | [[File:Subgroup of Oh; Dih4 blue orange 07; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 blue orange 16; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 blue orange 23; cycle graph.svg|200px|]] |- | [[File:Subgroup of Oh; Dih4 blue orange 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue orange 07; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 blue orange 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue orange 16; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 blue orange 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue orange 23; solid.png|80px|]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">D_2d</span> Dih₄ green orange <span style="float:right; padding-right:50px;">[2⁺,4]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 green orange 07; example solid.png|230px]] | [[File:Subgroup of Oh; Dih4 green orange 07; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 green orange 16; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 green orange 23; cycle graph.svg|200px|]] |- | [[File:Subgroup of Oh; Dih4 green orange 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green orange 07; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 green orange 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green orange 16; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 green orange 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 green orange 23; solid.png|80px|]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">D₄</span> Dih₄ blue red <span style="float:right; padding-right:50px;">[2,4]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; Dih4 blue red 07; example solid.png|230px]] | [[File:Subgroup of Oh; Dih4 blue red 07; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 blue red 16; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; Dih4 blue red 23; cycle graph.svg|200px|]] |- | [[File:Subgroup of Oh; Dih4 blue red 07; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue red 07; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 blue red 16; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue red 16; solid.png|80px|]] | [[File:Subgroup of Oh; Dih4 blue red 23; matrix.svg|50px|]] [[File:Subgroup of Oh; Dih4 blue red 23; solid.png|80px|]] |} <small>([[#Symmetry group of the square|Below]] the Dih₄ subgroups are shown in more detail.)</small> ====Subgroups of order 6==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="5"|<span style="float:left; padding-left:50px;">C_3v</span> S₃ green <span style="float:right; padding-right:50px;">[3]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S3 green 03; example solid.png|230px]] | [[File:Subgroup of Oh; S3 green 03; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; S3 green 11; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; S3 green 15; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; S3 green 08; cycle graph.svg|170px|]] |- | [[File:Subgroup of Oh; S3 green 03; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 03; solid.png|80px]] | [[File:Subgroup of Oh; S3 green 11; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 11; solid.png|80px]] | [[File:Subgroup of Oh; S3 green 15; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 15; solid.png|80px]] | [[File:Subgroup of Oh; S3 green 08; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 green 08; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="5"|<span style="float:left; padding-left:50px;">D₃</span> S₃ blue <span style="float:right; padding-right:50px;">[2,3]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; S3 blue 03; example solid.png|230px]] | [[File:Subgroup of Oh; S3 blue 03; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; S3 blue 11; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; S3 blue 15; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; S3 blue 08; cycle graph.svg|170px|]] |- | [[File:Subgroup of Oh; S3 blue 03; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 03; solid.png|80px]] | [[File:Subgroup of Oh; S3 blue 11; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 11; solid.png|80px]] | [[File:Subgroup of Oh; S3 blue 15; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 15; solid.png|80px]] | [[File:Subgroup of Oh; S3 blue 08; matrix.svg|50px|]] [[File:Subgroup of Oh; S3 blue 08; solid.png|80px]] |} <small>([[#Symmetry group of the triangle|Below]] the S₃ subgroups are shown in more detail.)</small> {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#e0e0e0" colspan="5"|<span style="float:left; padding-left:50px;">S₆</span> C₆ <span style="float:right; padding-right:50px;">[2⁺,6⁺]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C6 03; example solid.png|230px]] | [[File:Subgroup of Oh; C6 03; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; C6 11; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; C6 15; cycle graph.svg|200px|]] | [[File:Subgroup of Oh; C6 08; cycle graph.svg|200px|]] |- | [[File:Subgroup of Oh; C6 03; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 03; solid.png|80px]] | [[File:Subgroup of Oh; C6 11; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 11; solid.png|80px]] | [[File:Subgroup of Oh; C6 15; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 15; solid.png|80px]] | [[File:Subgroup of Oh; C6 08; matrix.svg|50px|]] [[File:Subgroup of Oh; C6 08; solid.png|80px]] |} ====Subgroups of order 4==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0e0e0" colspan="4"|<span style="float:left; padding-left:50px;">S₄</span> C₄ orange <span style="float:right; padding-right:50px;">[2⁺,4⁺]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C4 orange 07; example solid.png|230px]] | [[File:Subgroup of Oh; C4 orange 07; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; C4 orange 16; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; C4 orange 23; cycle graph.svg|180px|]] |- | [[File:Subgroup of Oh; C4 orange 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 orange 07; solid.png|80px]] | [[File:Subgroup of Oh; C4 orange 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 orange 16; solid.png|80px]] | [[File:Subgroup of Oh; C4 orange 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 orange 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">C₄</span> C₄ red <span style="float:right; padding-right:50px;">[4]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C4 red 07; example solid.png|230px]] | [[File:Subgroup of Oh; C4 red 07; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; C4 red 16; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; C4 red 23; cycle graph.svg|180px|]] |- | [[File:Subgroup of Oh; C4 red 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 red 07; solid.png|80px]] | [[File:Subgroup of Oh; C4 red 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 red 16; solid.png|80px]] | [[File:Subgroup of Oh; C4 red 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C4 red 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0f0e0" colspan="4"|<span style="float:left; padding-left:50px;">C_2h = D_1d</span> V inv white <span style="float:right; padding-right:50px;">[2⁺,2]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V inv white 07; example solid.png|230px]] | [[File:Subgroup of Oh; V inv white 07; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv white 16; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv white 23; cycle graph.svg|80px|]] |- | [[File:Subgroup of Oh; V inv white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V inv white 07; solid.png|80px]] | [[File:Subgroup of Oh; V inv white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V inv white 16; solid.png|80px]] | [[File:Subgroup of Oh; V inv white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V inv white 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0f0e0" colspan="7"|<span style="float:left; padding-left:50px;">C_2h = D_1d</span> V inv green <span style="float:right; padding-right:50px;">[2⁺,2]</span> |- |rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; V inv green 01; example solid.png|230px]] | [[File:Subgroup of Oh; V inv green 01; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv green 06; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv green 05; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv green 14; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv green 02; cycle graph.svg|80px|]] | [[File:Subgroup of Oh; V inv green 21; cycle graph.svg|80px|]] |- | [[File:Subgroup of Oh; V inv green 01; matrix.svg|50px|]] | [[File:Subgroup of Oh; V inv green 06; matrix.svg|50px|]] | [[File:Subgroup of Oh; V inv green 05; matrix.svg|50px|]] | [[File:Subgroup of Oh; V inv green 14; matrix.svg|50px|]] | [[File:Subgroup of Oh; V inv green 02; matrix.svg|50px|]] | [[File:Subgroup of Oh; V inv green 21; matrix.svg|50px|]] |- | [[File:Subgroup of Oh; V inv green 01; solid.png|80px]] | [[File:Subgroup of Oh; V inv green 06; solid.png|80px]] | [[File:Subgroup of Oh; V inv green 05; solid.png|80px]] | [[File:Subgroup of Oh; V inv green 14; solid.png|80px]] | [[File:Subgroup of Oh; V inv green 02; solid.png|80px]] | [[File:Subgroup of Oh; V inv green 21; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="7"|<span style="float:left; padding-left:50px;">C_2v</span> V green blue yellow <span style="float:right; padding-right:50px;">[2]</span> |- |rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; V gby 01; example solid.png|230px]] | [[File:Subgroup of Oh; V gby 01; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V gby 06; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V gby 05; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V gby 14; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V gby 02; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V gby 21; cycle graph.svg|180px|]] |- | [[File:Subgroup of Oh; V gby 01; matrix.svg|50px|]] | [[File:Subgroup of Oh; V gby 06; matrix.svg|50px|]] | [[File:Subgroup of Oh; V gby 05; matrix.svg|50px|]] | [[File:Subgroup of Oh; V gby 14; matrix.svg|50px|]] | [[File:Subgroup of Oh; V gby 02; matrix.svg|50px|]] | [[File:Subgroup of Oh; V gby 21; matrix.svg|50px|]] |- | [[File:Subgroup of Oh; V gby 01; solid.png|80px]] | [[File:Subgroup of Oh; V gby 06; solid.png|80px]] | [[File:Subgroup of Oh; V gby 05; solid.png|80px]] | [[File:Subgroup of Oh; V gby 14; solid.png|80px]] | [[File:Subgroup of Oh; V gby 02; solid.png|80px]] | [[File:Subgroup of Oh; V gby 21; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C_2v</span> V yellow white <span style="float:right; padding-right:50px;">[2]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V yellow white 07; example solid.png|230px]] | [[File:Subgroup of Oh; V yellow white 07; cycle graph.svg|150px|]] | [[File:Subgroup of Oh; V yellow white 16; cycle graph.svg|150px|]] | [[File:Subgroup of Oh; V yellow white 23; cycle graph.svg|150px|]] |- | [[File:Subgroup of Oh; V yellow white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V yellow white 07; solid.png|80px]] | [[File:Subgroup of Oh; V yellow white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V yellow white 16; solid.png|80px]] | [[File:Subgroup of Oh; V yellow white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V yellow white 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C_2v</span> V green white <span style="float:right; padding-right:50px;">[2]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V green white 07; example solid.png|230px]] | [[File:Subgroup of Oh; V green white 07; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V green white 16; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V green white 23; cycle graph.svg|180px|]] |- | [[File:Subgroup of Oh; V green white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V green white 07; solid.png|80px]] | [[File:Subgroup of Oh; V green white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V green white 16; solid.png|80px]] | [[File:Subgroup of Oh; V green white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V green white 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">D₂</span> V blue white <span style="float:right; padding-right:50px;">[2,2]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V blue white; example solid.png|230px]] | [[File:Subgroup of Oh; V blue white 07; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V blue white 16; cycle graph.svg|180px|]] | [[File:Subgroup of Oh; V blue white 23; cycle graph.svg|180px|]] |- | [[File:Subgroup of Oh; V blue white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; V blue white 07; solid.png|80px]] | [[File:Subgroup of Oh; V blue white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; V blue white 16; solid.png|80px]] | [[File:Subgroup of Oh; V blue white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; V blue white 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="2"|<span style="float:left; padding-left:50px;">D₂</span> V white <span style="float:right; padding-right:50px;">[2,2]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; V white; example solid.png|230px]] | [[File:Subgroup of Oh; V white; cycle graph.svg|220px|]] |- | [[File:Subgroup of Oh; V white; matrix.svg|50px|]] [[File:Subgroup of Oh; V white; solid.png|80px]] |} ====Subgroups of order 3==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="5"|<span style="float:left; padding-left:50px;">C₃</span> C₃ <span style="float:right; padding-right:50px;">[3]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C3 03; example solid.png|230px]] | [[File:Subgroup of Oh; C3 03; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; C3 11; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; C3 15; cycle graph.svg|170px|]] | [[File:Subgroup of Oh; C3 08; cycle graph.svg|170px|]] |- | [[File:Subgroup of Oh; C3 03; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 03; solid.png|80px]] | [[File:Subgroup of Oh; C3 11; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 11; solid.png|80px]] | [[File:Subgroup of Oh; C3 15; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 15; solid.png|80px]] | [[File:Subgroup of Oh; C3 08; matrix.svg|50px|]] [[File:Subgroup of Oh; C3 08; solid.png|80px]] |} ====Subgroups of order 2==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" !colspan="2" bgcolor="#e0e0e0"|<span style="float:left; padding-left:50px;">S₂</span> C₂ inv <span style="float:right; padding-right:50px;">[2⁺,2⁺]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2 inv; example solid.png|230px]] | [[File:Subgroup of Oh; C2 inv; cycle graph.svg|70px|]] |- | [[File:Subgroup of Oh; C2 inv; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 inv; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="4"|<span style="float:left; padding-left:50px;">C_s = C_1v</span> C₂ yellow <span style="float:right; padding-right:50px;">[ ]</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2 yellow 07; example solid.png|230px]] | [[File:Subgroup of Oh; C2 yellow 07; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 yellow 16; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 yellow 23; cycle graph.svg|70px|]] |- | [[File:Subgroup of Oh; C2 yellow 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 yellow 07; solid.png|80px]] | [[File:Subgroup of Oh; C2 yellow 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 yellow 16; solid.png|80px]] | [[File:Subgroup of Oh; C2 yellow 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 yellow 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="7"|<span style="float:left; padding-left:50px;">C_s = C_1v</span> C₂ green <span style="float:right; padding-right:50px;">[ ]</span> |- |rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; C2 green 01; example solid.png|230px]] | [[File:Subgroup of Oh; C2 green 01; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 green 06; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 green 05; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 green 14; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 green 02; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 green 21; cycle graph.svg|70px|]] |- | [[File:Subgroup of Oh; C2 green 01; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 green 06; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 green 05; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 green 14; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 green 02; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 green 21; matrix.svg|50px|]] |- | [[File:Subgroup of Oh; C2 green 01; solid.png|80px]] | [[File:Subgroup of Oh; C2 green 06; solid.png|80px]] | [[File:Subgroup of Oh; C2 green 05; solid.png|80px]] | [[File:Subgroup of Oh; C2 green 14; solid.png|80px]] | [[File:Subgroup of Oh; C2 green 02; solid.png|80px]] | [[File:Subgroup of Oh; C2 green 21; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0e0f0" colspan="4"|<span style="float:left; padding-left:50px;">C₂</span> C₂ white <span style="float:right; padding-right:50px;">[2]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; C2 white 07; example solid.png|230px]] | [[File:Subgroup of Oh; C2 white 07; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 white 16; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 white 23; cycle graph.svg|70px|]] |- | [[File:Subgroup of Oh; C2 white 07; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 white 07; solid.png|80px]] | [[File:Subgroup of Oh; C2 white 16; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 white 16; solid.png|80px]] | [[File:Subgroup of Oh; C2 white 23; matrix.svg|50px|]] [[File:Subgroup of Oh; C2 white 23; solid.png|80px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="7"|<span style="float:left; padding-left:50px;">C₂</span> C₂ blue <span style="float:right; padding-right:50px;">[2]⁺</span> |- |rowspan="3" style="width:240px;"| [[File:Subgroup of Oh; C2 blue 01; example solid.png|230px]] | [[File:Subgroup of Oh; C2 blue 01; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 blue 06; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 blue 05; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 blue 14; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 blue 02; cycle graph.svg|70px|]] | [[File:Subgroup of Oh; C2 blue 21; cycle graph.svg|70px|]] |- | [[File:Subgroup of Oh; C2 blue 01; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 blue 06; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 blue 05; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 blue 14; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 blue 02; matrix.svg|50px|]] | [[File:Subgroup of Oh; C2 blue 21; matrix.svg|50px|]] |- | [[File:Subgroup of Oh; C2 blue 01; solid.png|80px]] | [[File:Subgroup of Oh; C2 blue 06; solid.png|80px]] | [[File:Subgroup of Oh; C2 blue 05; solid.png|80px]] | [[File:Subgroup of Oh; C2 blue 14; solid.png|80px]] | [[File:Subgroup of Oh; C2 blue 02; solid.png|80px]] | [[File:Subgroup of Oh; C2 blue 21; solid.png|80px]] |} ====The trivial group==== {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="2"|<span style="float:left; padding-left:50px;">C₁</span> C₁ <span style="float:right; padding-right:50px;">[ ]⁺</span> |- |rowspan="2" style="width:240px;"| [[File:Subgroup of Oh; trivial; example solid.png|230px]] |style="padding:30px;"| [[File:Circlemarker.svg|45px|]] |- | [[File:Subgroup of Oh; trivial; matrix.svg|50px|]] [[File:Subgroup of Oh; trivial; solid.png|80px]] |} ===Different appearances of the same group=== ====Symmetry group of the cuboid==== The symmetry group of the cuboid C₂³ appears in two essentially different ways as a subgroup of O_h.<br> The one where the cuboid is the cube itself is the most intuitive one.<br> In the other one the cuboid is the original cube rotated by 45° around an axis. The one where it is rotated around the z-axis is shown below.<br> There are 4 individual subgroups <small>(see [[#Subgroups of order 8|above]]).</small> {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" !colspan="9" bgcolor="#e0f0f0"|<span style="float:left; padding-left:50px;">D_2h</span> C₂³ white <span style="float:right; padding-right:50px;">[2,2]</span> |- | [[File:Subgroup of Oh; C2^3 white; example solid.png|180px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 1 0.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 2 0.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 4 0.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 5 0.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 6 0.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 7 0.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#e0f0f0" colspan="9"|<span style="float:left; padding-left:50px;">D_2h</span> C₂³ green <span style="float:right; padding-right:50px;">[2,2]</span> |- | [[File:Subgroup of Oh; C2^3 green 07; example solid.png|180px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 3 1.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 4 0.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 4 1.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 7 0.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 0.svg|20px]] |} ====Symmetry group of the square==== The [[w:Dihedral group of order 8|symmetry group of the square]] appears in four essentially different ways as a subgroup of O_h. (C_4v or [4] being the most intuitive among them.)<br> There are 12 individual Dih₄ subgroups <small>(see [[#Subgroups of order 8|above]])</small>. Shown below are the four where the square is seen "from above", i.e. a point on the positive z-axis. In the white box above the colored boxes of the four subgroups are the permutations of the square. Their 2×2 transformation matrices are the top left submatrices of the four 3×3 matrices in the same column. So the last non-zero entry of the 3×3 matrix determines the permutation in this column. (So each column has only two different permutations.) The pattern of these eight last entries identifies the subgroup. It is shown on the left in the little 4×2 matrix under the example solid. {| style="width: 100%; text-align: center; border: 1px solid #fff;" | [[File:Blank300.png|180px]] | <div style="padding: 0 25px;">[[File:Square permutation fix.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation horz.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation vert.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation cross.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation desc.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation left.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation right.svg|100px]]</div> | <div style="padding: 0 25px;">[[File:Square permutation asc.svg|100px]]</div> |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="white" colspan="9"|Square permutations for comparison |- | [[File:Dihedral group of order 8; Cayley table (element orders 1,2,2,4,4,2,2,2); subgroup of S4 (left).svg|180px]] | [[File:Square permutation 0 0, colored.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Square permutation 1 0, colored.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Square permutation 2 0, colored.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Square permutation 3 0, colored.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Square permutation 0 1, colored.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Square permutation 1 1, colored.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Square permutation 2 1, colored.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Square permutation 3 1, colored.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="9"|<span style="float:left; padding-left:50px;">C_4v</span> Dih₄ green red <span style="float:right; padding-right:50px;">[4]</span> |- | [[File:Subgroup of Oh; Dih4 green red 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 0.svg|40px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 1 0.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 2 0.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 1 1.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 2 1.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 3 1.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0f0e0" colspan="9"|<span style="float:left; padding-left:50px;">D_2d</span> Dih₄ blue orange <span style="float:right; padding-right:50px;">[2⁺,4]</span> |- | [[File:Subgroup of Oh; Dih4 blue orange 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 1.svg|40px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 1 0.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 2 0.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 4 1.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 5 1.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 6 1.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#f0f0e0" colspan="9"|<span style="float:left; padding-left:50px;">D_2d</span> Dih₄ green orange <span style="float:right; padding-right:50px;">[2⁺,4]</span> |- | [[File:Subgroup of Oh; Dih4 green orange 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 2.svg|40px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 5 0.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 6 0.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 5 1.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 6 1.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 3 1.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="9"|<span style="float:left; padding-left:50px;">D₄</span> Dih₄ blue red <span style="float:right; padding-right:50px;">[2,4]⁺</span> |- | [[File:Subgroup of Oh; Dih4 blue red 07; example solid.png|180px]]<br>[[File:Square subgroup of the cube, pattern of bottom right digits in 3x3 matrices, 3.svg|40px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 5 0.svg|150px]]<br>[[File:Cube vertex number 5.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 6 0.svg|150px]]<br>[[File:Cube vertex number 6.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 3 0.svg|150px]]<br>[[File:Cube vertex number 3.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 4 1.svg|150px]]<br>[[File:Cube vertex number 4.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 1 1.svg|150px]]<br>[[File:Cube vertex number 1.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 2 1.svg|150px]]<br>[[File:Cube vertex number 2.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]] |} ====Symmetry group of the triangle==== The [[w:Dihedral group of order 6|symmetry group of the triangle]] appears in two essentially different ways as a subgroup of O_h, with C_3v or [3] being the most intuitive among them.<br> There are 8 individual subgroups <small>(see [[#Subgroups of order 6|above]])</small>. Shown below are the ones where the triangle is seen from a point on the negative main diagonal of the coordinate system.<br> {| style="width: 100%; text-align: center; border: 1px solid #fff;" | [[File:Blank300.png|180px]] | <div style="padding: 0 20px;">[[File:Triangle permutation fix.svg|110px]]</div> | <div style="padding: 0 20px;">[[File:Triangle permutation ref left.svg|110px]]</div> | <div style="padding: 0 20px;">[[File:Triangle permutation ref right.svg|110px]]</div> | <div style="padding: 0 20px;">[[File:Triangle permutation rot left.svg|110px]]</div> | <div style="padding: 0 20px;">[[File:Triangle permutation rot right.svg|110px]]</div> | <div style="padding: 0 20px;">[[File:Triangle permutation ref horz.svg|110px]]</div> |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999; border-bottom: 0;" ! bgcolor="#e0f0f0" colspan="7"|<span style="float:left; padding-left:50px;">C_3v</span> S₃ green <span style="float:right; padding-right:50px;">[3]</span> |- | [[File:Subgroup of Oh; S3 green 03; example solid (triangle).png|180px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 0 1.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 0 2.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 2.svg|20px]] | [[File:Cube permutation 0 3.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 3.svg|20px]] | [[File:Cube permutation 0 4.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 0 5.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 5.svg|20px]] |} {| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #999;" ! bgcolor="#f0e0f0" colspan="7"|<span style="float:left; padding-left:50px;">D₃</span> S₃ blue <span style="float:right; padding-right:50px;">[2,3]⁺</span> |- | [[File:Subgroup of Oh; S3 blue 03; example solid (with numbers).png|180px]] | [[File:Cube permutation 0 0.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 0.svg|20px]] | [[File:Cube permutation 7 1.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 1.svg|20px]] | [[File:Cube permutation 7 2.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 2.svg|20px]] | [[File:Cube permutation 0 3.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 3.svg|20px]] | [[File:Cube permutation 0 4.svg|150px]]<br>[[File:Cube vertex number 0.svg|20px]] [[File:Finite permutation number 4.svg|20px]] | [[File:Cube permutation 7 5.svg|150px]]<br>[[File:Cube vertex number 7.svg|20px]] [[File:Finite permutation number 5.svg|20px]] |} ===Cuboctahedral example solids and contained hexagons=== {| class="collapsible open" style="width: 100%; text-align: center; border: 1px solid #ddd;" ! bgcolor="#eee"| |- | {| class="wikitable" style="width: 100%; text-align: center;" ! |style="background: #f0e0f0;"| '''S₄ blue red''' |style="background: #f0f0e0;"| '''A₄ × C₂''' |style="background: #f0e0f0;"| '''A₄''' |style="background: #e0f0f0;"| '''S₄ green orange''' |- ! shown above |rowspan="2" | [[File:Subgroup of Oh; S4 blue red; example solid.png|200px]] | [[File:Subgroup of Oh; A4xC2; example solid.png|200px]] | [[File:Subgroup of Oh; A4; example solid.png|200px]] | [[File:Subgroup of Oh; S4 green orange; example solid.png|200px]] |- ! cuboctahedral | [[File:Subgroup of Oh; A4xC2; example solid (cuboctahedron).png|200px]] | [[File:Subgroup of Oh; A4; example solid (cuboctahedron).png|200px]] | [[File:Subgroup of Oh; S4 green orange; example solid (cuboctahedron).png|200px]] |- ! contained hexagon | [[File:Subgroup of Oh; S3 blue 03; example solid.png|200px]] | [[File:Subgroup of Oh; C6 03; example solid.png|200px]] | [[File:Subgroup of Oh; C3 03; example solid.png|200px]] | [[File:Subgroup of Oh; S3 green 03; example solid.png|200px]] |- ! |style="background: #f0e0f0;"| '''S₃ blue''' |style="background: #e0e0e0;"| '''C₆''' |style="background: #f0e0f0;"| '''C₃''' |style="background: #e0f0f0;"| '''S₃ green''' |} |} ==Code== The Python code used to create many of the illustrations in this article can be found here: https://github.com/watchduck/full_octahedral_group The following code shows bijections from pairs to other representations: {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;" !style="background-color: #eee; font-weight: normal;"| '''S₄'''-based identifiers |- | <math>n'</math> is represented as <math>n+24</math>. <span style="color:gray;">(0, 0):</span> 0, <span style="color:gray;">(0, 1):</span> 1, <span style="color:gray;">(0, 2):</span> 2, <span style="color:gray;">(0, 3):</span> 3, <span style="color:gray;">(0, 4):</span> 4, <span style="color:gray;">(0, 5):</span> 5, <span style="color:gray;">(1, 0):</span> 47, <span style="color:gray;">(1, 1):</span> 46, <span style="color:gray;">(1, 2):</span> 45, <span style="color:gray;">(1, 3):</span> 44, <span style="color:gray;">(1, 4):</span> 43, <span style="color:gray;">(1, 5):</span> 42, <span style="color:gray;">(2, 0):</span> 40, <span style="color:gray;">(2, 1):</span> 41, <span style="color:gray;">(2, 2):</span> 37, <span style="color:gray;">(2, 3):</span> 36, <span style="color:gray;">(2, 4):</span> 39, <span style="color:gray;">(2, 5):</span> 38, <span style="color:gray;">(3, 0):</span> 7, <span style="color:gray;">(3, 1):</span> 6, <span style="color:gray;">(3, 2):</span> 10, <span style="color:gray;">(3, 3):</span> 11, <span style="color:gray;">(3, 4):</span> 8, <span style="color:gray;">(3, 5):</span> 9, <span style="color:gray;">(4, 0):</span> 31, <span style="color:gray;">(4, 1):</span> 30, <span style="color:gray;">(4, 2):</span> 34, <span style="color:gray;">(4, 3):</span> 35, <span style="color:gray;">(4, 4):</span> 32, <span style="color:gray;">(4, 5):</span> 33, <span style="color:gray;">(5, 0):</span> 16, <span style="color:gray;">(5, 1):</span> 17, <span style="color:gray;">(5, 2):</span> 13, <span style="color:gray;">(5, 3):</span> 12, <span style="color:gray;">(5, 4):</span> 15, <span style="color:gray;">(5, 5):</span> 14, <span style="color:gray;">(6, 0):</span> 23, <span style="color:gray;">(6, 1):</span> 22, <span style="color:gray;">(6, 2):</span> 21, <span style="color:gray;">(6, 3):</span> 20, <span style="color:gray;">(6, 4):</span> 19, <span style="color:gray;">(6, 5):</span> 18, <span style="color:gray;">(7, 0):</span> 24, <span style="color:gray;">(7, 1):</span> 25, <span style="color:gray;">(7, 2):</span> 26, <span style="color:gray;">(7, 3):</span> 27, <span style="color:gray;">(7, 4):</span> 28, <span style="color:gray;">(7, 5):</span> 29 |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;" !style="background-color: #eee; font-weight: normal;"| Permutations of '''cube''' vertices |- | The initial order is <code>(-1, -1, -1), (1, -1, -1), (-1, 1, -1), (1, 1, -1), (-1, -1, 1), (1, -1, 1), (-1, 1, 1), (1, 1, 1)</code>. <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> (0, 1, 2, 3, 4, 5, 6, 7), <span style="color:gray;">(0, 1):</span> (0, 2, 1, 3, 4, 6, 5, 7), <span style="color:gray;">(0, 2):</span> (0, 1, 4, 5, 2, 3, 6, 7), <span style="color:gray;">(0, 3):</span> (0, 4, 1, 5, 2, 6, 3, 7), <span style="color:gray;">(0, 4):</span> (0, 2, 4, 6, 1, 3, 5, 7), <span style="color:gray;">(0, 5):</span> (0, 4, 2, 6, 1, 5, 3, 7),</span> <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> (1, 0, 3, 2, 5, 4, 7, 6), <span style="color:gray;">(1, 1):</span> (1, 3, 0, 2, 5, 7, 4, 6), <span style="color:gray;">(1, 2):</span> (1, 0, 5, 4, 3, 2, 7, 6), <span style="color:gray;">(1, 3):</span> (1, 5, 0, 4, 3, 7, 2, 6), <span style="color:gray;">(1, 4):</span> (1, 3, 5, 7, 0, 2, 4, 6), <span style="color:gray;">(1, 5):</span> (1, 5, 3, 7, 0, 4, 2, 6),</span> <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> (2, 3, 0, 1, 6, 7, 4, 5), <span style="color:gray;">(2, 1):</span> (2, 0, 3, 1, 6, 4, 7, 5), <span style="color:gray;">(2, 2):</span> (2, 3, 6, 7, 0, 1, 4, 5), <span style="color:gray;">(2, 3):</span> (2, 6, 3, 7, 0, 4, 1, 5), <span style="color:gray;">(2, 4):</span> (2, 0, 6, 4, 3, 1, 7, 5), <span style="color:gray;">(2, 5):</span> (2, 6, 0, 4, 3, 7, 1, 5),</span> <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> (3, 2, 1, 0, 7, 6, 5, 4), <span style="color:gray;">(3, 1):</span> (3, 1, 2, 0, 7, 5, 6, 4), <span style="color:gray;">(3, 2):</span> (3, 2, 7, 6, 1, 0, 5, 4), <span style="color:gray;">(3, 3):</span> (3, 7, 2, 6, 1, 5, 0, 4), <span style="color:gray;">(3, 4):</span> (3, 1, 7, 5, 2, 0, 6, 4), <span style="color:gray;">(3, 5):</span> (3, 7, 1, 5, 2, 6, 0, 4),</span> <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> (4, 5, 6, 7, 0, 1, 2, 3), <span style="color:gray;">(4, 1):</span> (4, 6, 5, 7, 0, 2, 1, 3), <span style="color:gray;">(4, 2):</span> (4, 5, 0, 1, 6, 7, 2, 3), <span style="color:gray;">(4, 3):</span> (4, 0, 5, 1, 6, 2, 7, 3), <span style="color:gray;">(4, 4):</span> (4, 6, 0, 2, 5, 7, 1, 3), <span style="color:gray;">(4, 5):</span> (4, 0, 6, 2, 5, 1, 7, 3),</span> <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> (5, 4, 7, 6, 1, 0, 3, 2), <span style="color:gray;">(5, 1):</span> (5, 7, 4, 6, 1, 3, 0, 2), <span style="color:gray;">(5, 2):</span> (5, 4, 1, 0, 7, 6, 3, 2), <span style="color:gray;">(5, 3):</span> (5, 1, 4, 0, 7, 3, 6, 2), <span style="color:gray;">(5, 4):</span> (5, 7, 1, 3, 4, 6, 0, 2), <span style="color:gray;">(5, 5):</span> (5, 1, 7, 3, 4, 0, 6, 2),</span> <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> (6, 7, 4, 5, 2, 3, 0, 1), <span style="color:gray;">(6, 1):</span> (6, 4, 7, 5, 2, 0, 3, 1), <span style="color:gray;">(6, 2):</span> (6, 7, 2, 3, 4, 5, 0, 1), <span style="color:gray;">(6, 3):</span> (6, 2, 7, 3, 4, 0, 5, 1), <span style="color:gray;">(6, 4):</span> (6, 4, 2, 0, 7, 5, 3, 1), <span style="color:gray;">(6, 5):</span> (6, 2, 4, 0, 7, 3, 5, 1),</span> <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> (7, 6, 5, 4, 3, 2, 1, 0), <span style="color:gray;">(7, 1):</span> (7, 5, 6, 4, 3, 1, 2, 0), <span style="color:gray;">(7, 2):</span> (7, 6, 3, 2, 5, 4, 1, 0), <span style="color:gray;">(7, 3):</span> (7, 3, 6, 2, 5, 1, 4, 0), <span style="color:gray;">(7, 4):</span> (7, 5, 3, 1, 6, 4, 2, 0), <span style="color:gray;">(7, 5):</span> (7, 3, 5, 1, 6, 2, 4, 0)</span> |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;" !style="background-color: #eee; font-weight: normal;"| Permutations of '''octahedron''' vertices |- | The initial order is <code>(-1, 0, 0), (1, 0, 0), (0, -1, 0), (0, 1, 0), (0, 0, -1), (0, 0, 1)</code>. <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> (0, 1, 2, 3, 4, 5), <span style="color:gray;">(0, 1):</span> (2, 3, 0, 1, 4, 5), <span style="color:gray;">(0, 2):</span> (0, 1, 4, 5, 2, 3), <span style="color:gray;">(0, 3):</span> (4, 5, 0, 1, 2, 3), <span style="color:gray;">(0, 4):</span> (2, 3, 4, 5, 0, 1), <span style="color:gray;">(0, 5):</span> (4, 5, 2, 3, 0, 1),</span> <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> (1, 0, 2, 3, 4, 5), <span style="color:gray;">(1, 1):</span> (2, 3, 1, 0, 4, 5), <span style="color:gray;">(1, 2):</span> (1, 0, 4, 5, 2, 3), <span style="color:gray;">(1, 3):</span> (4, 5, 1, 0, 2, 3), <span style="color:gray;">(1, 4):</span> (2, 3, 4, 5, 1, 0), <span style="color:gray;">(1, 5):</span> (4, 5, 2, 3, 1, 0),</span> <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> (0, 1, 3, 2, 4, 5), <span style="color:gray;">(2, 1):</span> (3, 2, 0, 1, 4, 5), <span style="color:gray;">(2, 2):</span> (0, 1, 4, 5, 3, 2), <span style="color:gray;">(2, 3):</span> (4, 5, 0, 1, 3, 2), <span style="color:gray;">(2, 4):</span> (3, 2, 4, 5, 0, 1), <span style="color:gray;">(2, 5):</span> (4, 5, 3, 2, 0, 1),</span> <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> (1, 0, 3, 2, 4, 5), <span style="color:gray;">(3, 1):</span> (3, 2, 1, 0, 4, 5), <span style="color:gray;">(3, 2):</span> (1, 0, 4, 5, 3, 2), <span style="color:gray;">(3, 3):</span> (4, 5, 1, 0, 3, 2), <span style="color:gray;">(3, 4):</span> (3, 2, 4, 5, 1, 0), <span style="color:gray;">(3, 5):</span> (4, 5, 3, 2, 1, 0),</span> <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> (0, 1, 2, 3, 5, 4), <span style="color:gray;">(4, 1):</span> (2, 3, 0, 1, 5, 4), <span style="color:gray;">(4, 2):</span> (0, 1, 5, 4, 2, 3), <span style="color:gray;">(4, 3):</span> (5, 4, 0, 1, 2, 3), <span style="color:gray;">(4, 4):</span> (2, 3, 5, 4, 0, 1), <span style="color:gray;">(4, 5):</span> (5, 4, 2, 3, 0, 1),</span> <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> (1, 0, 2, 3, 5, 4), <span style="color:gray;">(5, 1):</span> (2, 3, 1, 0, 5, 4), <span style="color:gray;">(5, 2):</span> (1, 0, 5, 4, 2, 3), <span style="color:gray;">(5, 3):</span> (5, 4, 1, 0, 2, 3), <span style="color:gray;">(5, 4):</span> (2, 3, 5, 4, 1, 0), <span style="color:gray;">(5, 5):</span> (5, 4, 2, 3, 1, 0),</span> <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> (0, 1, 3, 2, 5, 4), <span style="color:gray;">(6, 1):</span> (3, 2, 0, 1, 5, 4), <span style="color:gray;">(6, 2):</span> (0, 1, 5, 4, 3, 2), <span style="color:gray;">(6, 3):</span> (5, 4, 0, 1, 3, 2), <span style="color:gray;">(6, 4):</span> (3, 2, 5, 4, 0, 1), <span style="color:gray;">(6, 5):</span> (5, 4, 3, 2, 0, 1),</span> <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> (1, 0, 3, 2, 5, 4), <span style="color:gray;">(7, 1):</span> (3, 2, 1, 0, 5, 4), <span style="color:gray;">(7, 2):</span> (1, 0, 5, 4, 3, 2), <span style="color:gray;">(7, 3):</span> (5, 4, 1, 0, 3, 2), <span style="color:gray;">(7, 4):</span> (3, 2, 5, 4, 1, 0), <span style="color:gray;">(7, 5):</span> (5, 4, 3, 2, 1, 0)</span> |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;" !colspan="2" style="background-color: #eee; font-weight: normal;"| Pairs in '''reverse colex order''' |- |colspan="2"| The [[Lexicographic_and_colexicographic_order#Permutations|reverse colexicographic order of permutations]] is the order of their respective {{w|Factorial number system|integer values}}.<br> The list of permutations of octahedron vertices on the right can be generalized as the infinite list of {{w|cross-polytope}} vertices.<br> <small>(Its first eight elements are the permutations of the square within a 4×2 matrix.)</small> |- style="text-align: center;" | '''cube''' | '''octahedron''' |- style="vertical-align: top;" | (0, 0), (0, 2), (0, 1), (0, 4), (0, 3), (0, 5), (1, 0), (1, 2), (1, 1), (1, 4), (1, 3), (1, 5), (2, 1), (2, 4), (2, 0), (2, 2), (2, 5), (2, 3), (3, 1), (3, 4), (3, 0), (3, 2), (3, 5), (3, 3), (4, 3), (4, 5), (4, 2), (4, 0), (4, 4), (4, 1), (5, 3), (5, 5), (5, 2), (5, 0), (5, 4), (5, 1), (6, 5), (6, 3), (6, 4), (6, 1), (6, 2), (6, 0), (7, 5), (7, 3), (7, 4), (7, 1), (7, 2), (7, 0) | (0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (2, 1), (1, 1), (3, 1), (4, 0), (5, 0), (6, 0), (7, 0), (4, 1), (6, 1), (5, 1), (7, 1), (0, 2), (1, 2), (4, 2), (5, 2), (0, 3), (4, 3), (1, 3), (5, 3), (2, 2), (3, 2), (6, 2), (7, 2), (2, 3), (6, 3), (3, 3), (7, 3), (0, 4), (2, 4), (4, 4), (6, 4), (0, 5), (4, 5), (2, 5), (6, 5), (1, 4), (3, 4), (5, 4), (7, 4), (1, 5), (5, 5), (3, 5), (7, 5) |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; border-bottom: none; background-color: white;" !style="background-color: #eee; font-weight: normal;"| '''3×3 matrices''' |- | <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> [[ 1, 0, 0], [0, 1, 0], [0, 0, 1]], <span style="color:gray;">(0, 1):</span> [[0, 1, 0], [ 1, 0, 0], [0, 0, 1]], <span style="color:gray;">(0, 2):</span> [[ 1, 0, 0], [0, 0, 1], [0, 1, 0]], <span style="color:gray;">(0, 3):</span> [[0, 1, 0], [0, 0, 1], [ 1, 0, 0]], <span style="color:gray;">(0, 4):</span> [[0, 0, 1], [ 1, 0, 0], [0, 1, 0]], <span style="color:gray;">(0, 5):</span> [[0, 0, 1], [0, 1, 0], [ 1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> [[-1, 0, 0], [0, 1, 0], [0, 0, 1]], <span style="color:gray;">(1, 1):</span> [[0,-1, 0], [ 1, 0, 0], [0, 0, 1]], <span style="color:gray;">(1, 2):</span> [[-1, 0, 0], [0, 0, 1], [0, 1, 0]], <span style="color:gray;">(1, 3):</span> [[0,-1, 0], [0, 0, 1], [ 1, 0, 0]], <span style="color:gray;">(1, 4):</span> [[0, 0,-1], [ 1, 0, 0], [0, 1, 0]], <span style="color:gray;">(1, 5):</span> [[0, 0,-1], [0, 1, 0], [ 1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> [[ 1, 0, 0], [0,-1, 0], [0, 0, 1]], <span style="color:gray;">(2, 1):</span> [[0, 1, 0], [-1, 0, 0], [0, 0, 1]], <span style="color:gray;">(2, 2):</span> [[ 1, 0, 0], [0, 0,-1], [0, 1, 0]], <span style="color:gray;">(2, 3):</span> [[0, 1, 0], [0, 0,-1], [ 1, 0, 0]], <span style="color:gray;">(2, 4):</span> [[0, 0, 1], [-1, 0, 0], [0, 1, 0]], <span style="color:gray;">(2, 5):</span> [[0, 0, 1], [0,-1, 0], [ 1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> [[-1, 0, 0], [0,-1, 0], [0, 0, 1]], <span style="color:gray;">(3, 1):</span> [[0,-1, 0], [-1, 0, 0], [0, 0, 1]], <span style="color:gray;">(3, 2):</span> [[-1, 0, 0], [0, 0,-1], [0, 1, 0]], <span style="color:gray;">(3, 3):</span> [[0,-1, 0], [0, 0,-1], [ 1, 0, 0]], <span style="color:gray;">(3, 4):</span> [[0, 0,-1], [-1, 0, 0], [0, 1, 0]], <span style="color:gray;">(3, 5):</span> [[0, 0,-1], [0,-1, 0], [ 1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> [[ 1, 0, 0], [0, 1, 0], [0, 0,-1]], <span style="color:gray;">(4, 1):</span> [[0, 1, 0], [ 1, 0, 0], [0, 0,-1]], <span style="color:gray;">(4, 2):</span> [[ 1, 0, 0], [0, 0, 1], [0,-1, 0]], <span style="color:gray;">(4, 3):</span> [[0, 1, 0], [0, 0, 1], [-1, 0, 0]], <span style="color:gray;">(4, 4):</span> [[0, 0, 1], [ 1, 0, 0], [0,-1, 0]], <span style="color:gray;">(4, 5):</span> [[0, 0, 1], [0, 1, 0], [-1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> [[-1, 0, 0], [0, 1, 0], [0, 0,-1]], <span style="color:gray;">(5, 1):</span> [[0,-1, 0], [ 1, 0, 0], [0, 0,-1]], <span style="color:gray;">(5, 2):</span> [[-1, 0, 0], [0, 0, 1], [0,-1, 0]], <span style="color:gray;">(5, 3):</span> [[0,-1, 0], [0, 0, 1], [-1, 0, 0]], <span style="color:gray;">(5, 4):</span> [[0, 0,-1], [ 1, 0, 0], [0,-1, 0]], <span style="color:gray;">(5, 5):</span> [[0, 0,-1], [0, 1, 0], [-1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> [[ 1, 0, 0], [0,-1, 0], [0, 0,-1]], <span style="color:gray;">(6, 1):</span> [[0, 1, 0], [-1, 0, 0], [0, 0,-1]], <span style="color:gray;">(6, 2):</span> [[ 1, 0, 0], [0, 0,-1], [0,-1, 0]], <span style="color:gray;">(6, 3):</span> [[0, 1, 0], [0, 0,-1], [-1, 0, 0]], <span style="color:gray;">(6, 4):</span> [[0, 0, 1], [-1, 0, 0], [0,-1, 0]], <span style="color:gray;">(6, 5):</span> [[0, 0, 1], [0,-1, 0], [-1, 0, 0]],</span> <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> [[-1, 0, 0], [0,-1, 0], [0, 0,-1]], <span style="color:gray;">(7, 1):</span> [[0,-1, 0], [-1, 0, 0], [0, 0,-1]], <span style="color:gray;">(7, 2):</span> [[-1, 0, 0], [0, 0,-1], [0,-1, 0]], <span style="color:gray;">(7, 3):</span> [[0,-1, 0], [0, 0,-1], [-1, 0, 0]], <span style="color:gray;">(7, 4):</span> [[0, 0,-1], [-1, 0, 0], [0,-1, 0]], <span style="color:gray;">(7, 5):</span> [[0, 0,-1], [0,-1, 0], [-1, 0, 0]]</span> |} {| class="collapsible collapsed" style="width: 100%; border: 1px solid #ddd; background-color: white;" !style="background-color: #eee; font-weight: normal;"| '''Truncated cuboctahedron''' coordinates |- | The points of the {{w|truncated cuboctahedron}} are the <math>2^3 \cdot 6! = 48</math> permutations of <math>\bigl(\pm 1, \pm (\sqrt{2} + 1), \pm (2 \sqrt{2} + 1)\bigr)</math>.<br> E.g. <math>(5, 4) = (1, -b, a) = (1, -2 \sqrt{2} -1, \sqrt{2} +1)</math>. a = sqrt(2) + 1 b = 2*sqrt(2) + 1 <span style="white-space: nowrap;"><span style="color:gray;">(0, 0):</span> (-b, -a, -1), <span style="color:gray;">(0, 1):</span> (-a, -b, -1), <span style="color:gray;">(0, 2):</span> (-b, -1, -a), <span style="color:gray;">(0, 3):</span> (-a, -1, -b), <span style="color:gray;">(0, 4):</span> (-1, -b, -a), <span style="color:gray;">(0, 5):</span> (-1, -a, -b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(1, 0):</span> ( b, -a, -1), <span style="color:gray;">(1, 1):</span> ( a, -b, -1), <span style="color:gray;">(1, 2):</span> ( b, -1, -a), <span style="color:gray;">(1, 3):</span> ( a, -1, -b), <span style="color:gray;">(1, 4):</span> ( 1, -b, -a), <span style="color:gray;">(1, 5):</span> ( 1, -a, -b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(2, 0):</span> (-b, &nbsp;a, -1), <span style="color:gray;">(2, 1):</span> (-a, &nbsp;b, -1), <span style="color:gray;">(2, 2):</span> (-b, &nbsp;1, -a), <span style="color:gray;">(2, 3):</span> (-a, &nbsp;1, -b), <span style="color:gray;">(2, 4):</span> (-1, &nbsp;b, -a), <span style="color:gray;">(2, 5):</span> (-1, &nbsp;a, -b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(3, 0):</span> ( b, &nbsp;a, -1), <span style="color:gray;">(3, 1):</span> ( a, &nbsp;b, -1), <span style="color:gray;">(3, 2):</span> ( b, &nbsp;1, -a), <span style="color:gray;">(3, 3):</span> ( a, &nbsp;1, -b), <span style="color:gray;">(3, 4):</span> ( 1, &nbsp;b, -a), <span style="color:gray;">(3, 5):</span> ( 1, &nbsp;a, -b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(4, 0):</span> (-b, -a, &nbsp;1), <span style="color:gray;">(4, 1):</span> (-a, -b, &nbsp;1), <span style="color:gray;">(4, 2):</span> (-b, -1, &nbsp;a), <span style="color:gray;">(4, 3):</span> (-a, -1, &nbsp;b), <span style="color:gray;">(4, 4):</span> (-1, -b, &nbsp;a), <span style="color:gray;">(4, 5):</span> (-1, -a, &nbsp;b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(5, 0):</span> ( b, -a, &nbsp;1), <span style="color:gray;">(5, 1):</span> ( a, -b, &nbsp;1), <span style="color:gray;">(5, 2):</span> ( b, -1, &nbsp;a), <span style="color:gray;">(5, 3):</span> ( a, -1, &nbsp;b), <span style="color:gray;">(5, 4):</span> ( 1, -b, &nbsp;a), <span style="color:gray;">(5, 5):</span> ( 1, -a, &nbsp;b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(6, 0):</span> (-b, &nbsp;a, &nbsp;1), <span style="color:gray;">(6, 1):</span> (-a, &nbsp;b, &nbsp;1), <span style="color:gray;">(6, 2):</span> (-b, &nbsp;1, &nbsp;a), <span style="color:gray;">(6, 3):</span> (-a, &nbsp;1, &nbsp;b), <span style="color:gray;">(6, 4):</span> (-1, &nbsp;b, &nbsp;a), <span style="color:gray;">(6, 5):</span> (-1, &nbsp;a, &nbsp;b),</span> <span style="white-space: nowrap;"><span style="color:gray;">(7, 0):</span> ( b, &nbsp;a, &nbsp;1), <span style="color:gray;">(7, 1):</span> ( a, &nbsp;b, &nbsp;1), <span style="color:gray;">(7, 2):</span> ( b, &nbsp;1, &nbsp;a), <span style="color:gray;">(7, 3):</span> ( a, &nbsp;1, &nbsp;b), <span style="color:gray;">(7, 4):</span> ( 1, &nbsp;b, &nbsp;a), <span style="color:gray;">(7, 5):</span> ( 1, &nbsp;a, &nbsp;b)</span> |} <small>These are Python dictionaries without the surrounding braces. (Dicts work only in one direction, but [https://pypi.python.org/pypi/bidict bidict] can be used to get back to the pairs.)</small> A dictionary of these permutations and their properties (including conjugacy class and inverse) can be found [https://pastebin.com/raw/JBx0TSsC here]. A dictionary of all the subgroups can be found [https://github.com/watchduck/full_octahedral_group/blob/master/projects/p03_subgroups/store_dicts.py here] (as a bijection from naive names to tuples of S₄ based numbers). [[Category:Full octahedral group]] gyykf9jdxalgs3of4rgif6kmher63ic 14 217068 2816051 2557222 2026-06-17T00:29:43Z Jtneill 10242 removed [[Category:Health]]; added [[Category:Health psychology]] using [[Help:Gadget-HotCat|HotCat]] 2816051 wikitext text/x-wiki [[Category:{{#titleparts:{{PAGENAME}}|2}}]] [[Category:Health psychology]] 5gq0s9nh84g9bs0pj3g7nxu554n8mec 0 221601 2816064 2815367 2026-06-17T05:37:36Z Jtneill 10242 /* References (10%) */ (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) 2816064 wikitext text/x-wiki {{title|Topic development — Guidelines}} <div style="text-align: center;">''Chapter plan and user page'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 08 |day = 14 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester -->{{:Motivation and emotion/Assessment/Chapter/Contents}}</div> {{TOCright}} ==Overview== * Weight: 10% * Due: {{/Due}} * Tasks ** Create a Wikiversity user account ** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]] ** Build wiki editing skills by developing a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] which consists of: *** Title and sub-title *** Headings (and possibly sub-headings) **** Overview **** 3-5 other top-level headings **** Conclusion **** See also (with at least 2 lins (1 Wikiveristy and 1 Wikipedia)) **** References (at least 6) **** External links (at least 2) *** Key points for each section (and sub-section) *** Figure (at least 1) *** Learning feature (plan at least 1) ** Create a Wikiversity user page *** Introduce yourself *** Summarise at least three different types of social contributions on your Wikiversity user page * Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]] * Guidance for this assignment is provided in Module 1: ** [[Motivation and emotion/Lectures/Introduction|Lecture 01]] ** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] *Feedback will be provided to help guide drafting of the full [[Motivation and emotion/Assessment/Chapter|book chapter]] *Marks and feedback should be returned before Census Date (end of Week 4) **Marks will be available via {{Motivation and emotion/Canvas}} **Written feedback will be available via the topic's Wikiversity discussion page *Follow up if you don't understand or would like more feedback ==Extensions and late submissions== * Extension requests require an Extension Application Form to be submitted via {{Motivation and emotion/Canvas}} with appropriate documentary evidence * Submissions are accepted up to 3 days late (-10% per day late) * If you don't submit this assessment on time, withdrawal from the unit before Census Date (end of Week 4) is recommended ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic. |- style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Be professional — communicate effectively | Communicate your ideas by sharing a chapter plan; provide feedback on other plans. |- style="vertical-align:top;" | Be professional — display initiative and drive, and use organisation skills to plan and manage workload | Get organised by selecting a topic and submitting an on-time chapter plan. |- style="vertical-align:top;" | Be a lifelong learner — evaluate and adopt new technology | Learn how to edit in a collaborative, online environment. |} ==Instructions== Follow these instructions for the topic development: * Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of: *# Title and sub-title (pre-approved or negotiated) *# Overview *# 3-5 other top-level headings *# Key points for each heading/sub-heading with citations *# 1+ relevant figure(s) *# 1+ actual or planned learning feature *# 6+ references *# 4+ resources *#* See also: 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article) *#* External links: 2+ external links (to external resources) *# Wikiversity user page *#* self-introduction which links to the chapter being worked on *#* Social contributions in a numbered list with a summary and direct link to evidence *#** 1 direct edit to improve another book chapter (past or present) *#** 1 talk page comment on another book chapter (past or present) *#** 1 {{Motivation and emotion/Canvas}} discussion post * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * <span id="Word count">Length (Word count):</span> There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700). * Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments ==Template== {{:Motivation and emotion/Assessment/Topic/Quickstarttip}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] {{anchor|Title}} ===Title and sub-title (10%)=== * Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}}) * Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] * Do not include user name; authorship is as per the page's editing history {{anchor|Headings}} ===Headings (10%)=== * Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links) * Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading. * The top-level headings should align with the sub-title and focus questions * Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]]) {{anchor|Overview}} ===Overview (10%)=== * A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] * At least 3 bullet points outlining the "problem" (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]] * 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] {{anchor|Key points}} ===Key points (10%)=== * At least 3 bullet points per section (i.e., per heading or sub-heading) * Overview the most relevant theory(ies), including key citations * Overview the most relevant research, including key citations * Provide at least 1 introductory bullet point before branching into sub-sections * Address the problem (i.e., answer the question in the sub-title) {{Anchor|Figure}} ===Figure (10%)=== * Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]]. * Number each figure sequentially (e.g., Figure 1, Figure 2 etc.) * Include a descriptive caption that connects the figure to the text * Cite each figure at least once in the main text (e.g., see Figure 1) * Optimise image display size to make it easy to read (i.e., not too big or too small) {{Anchor|Learning feature}} ===Learning feature (10%)=== * In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,: ** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] ** Internal (wiki) links: *** At least 1 embedded link to a relevant book chapter *** At least 1 embedded link to a relevant Wikipedia article * Quiz question with correct and incorrect answers ** Table with an APA style caption {{anchor|References}} ===References (10%)=== * Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals) * Each source should be cited at least once in the key points * Include a balance of key theoretical and key research articles {{anchor|Resources}} ===Resources (10%)=== * '''See also''' (heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article) ** Provide at least 1 bullet-pointed: *** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter *** internal wiki link to a relevant Wikipedia page ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., Book chapter, 2023) ** Use alphabetical order * '''External links''' (heading): Provide at least 2 external links to key internet resources ** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles) ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., The Conversation) ** Use alphabetical order {{anchor|User page}} ===User page (10%)=== * Create a Wikiversity user page for your user account * Edit the user page to provide information about yourself * Recommended headings: ** About me ** Book chapter I'm working on *** Include an internal (wiki) link to the chapter page ** Social contributions * Consider linking to your other online profiles {{anchor|Social contribution}} {{anchor|Socialcontribution}} ===Social contribution (10%)=== * On your Wikiversity user page, summarise and link to direct evidence that you have made at least 3 different types of contributions: ** direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics) ** provide useful feedback by commenting on a book chapter's talk page (current or previous topic talk pages) ** post to the {{Motivation and emotion/Canvas}} discussion forum<!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --> * [[Motivation and emotion/Wikiversity/Social contributions|More info]] ==Grade descriptions== This section describes typical characteristics of topic developments at each grade level, based on the [[#Marking criteria|marking criteria]]. {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | A clear, complete, easy to understand plan is presented. Considerable depth and breadth of theoretical and research knowledge of the topic is demonstrated via the scope and detail within the plan. All recommended sections are provided. The development of the plan illustrates that the author has actively engaged in developing skills required for collaborative online writing and editing (e.g., interwiki links are provided for key terms, responses are made to comments on the chapter talk page). There are citations to more than 6 key academic sources with references provided in APA style. The author introduces themself on their Wikiversity user page and summarises and provides directly verifiable evidence of editing another chapter, comment provided on another chapter's talk page, and posting to the discussion forum. |- | style="vertical-align:top;" | '''DI (Distinction)''' | A very good, understandable plan is presented. The plan includes key relevant theory and research, with relevant references. The material is well organised into sections, with minimal spelling and grammar issues. There is good evidence that the author has developed the capacity to work effectively in the collaborative editing environment. The author's user page is set up and links to evidence of social contributions. However, there is at least 1 area for improvement. |- | style="vertical-align:top;" | '''CR (Credit)''' | A competent plan is presented. The plan includes the main ideas and sections necessary for developing a good chapter about the topic. Some aspects of the plan, however, may be missing, limited, or problematic. For example, the headings and structure may be under-developed, the reference list may indicate a lack of depth in investigation of the topic, use of wiki links and/or images could often be improved, and/or user page set-up feedback about other chapters may not have been completed. |- | style="vertical-align:top;" | '''P (Pass)''' | A basic, sufficient plan is presented, however there may be incomplete coverage of relevant theory and research, and/or a lack of depth or breadth in conceptualising the chapter. The chapter plan covers basic theory and research about the topic, but lacks detail about how the concepts will be brought together to help address the topic. A basic heading structure is presented, but is likely to need more sections and/or improved formatting or organisation. Spelling and grammar problems are often evident. Citation and referencing tends to be missing or limited in scope and quality (e.g., top peer-reviewed citations about the topic haven't been cited). These plans usually have very brief edit histories (e.g., less than 24 hours) and are often noticeably shorter than plans which attract higher grades. Authors often haven't set up an informative user page or provided evidence of engagement with the development of other chapter plans. |- | style="vertical-align:top;" | '''F (Fail)''' | The plan is insufficient and/or incomplete. Major gaps and/or errors in content are evident. Little evidence of awareness of relevant theory, research, and use of peer-reviewed references. These plans typically have under-developed heading structures and do not illustrate the use of key editing skills. Written expression is often undermined by poor spelling and/or grammar. These plans typically have very brief editing histories (e.g., consist of a few, last minute edits). There is generally no evidence of active engagement with the development of other chapters. |} ==Examples== ;About * Below are some examples of topic development submissions which received 100% * The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions * It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter) ;2025 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&oldid=2726043 Elina.jean.r] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&oldid=2727975 U3254978] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&oldid=2739659 U3237996] ;2024 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&oldid=2644098 U3216883] ;2023 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&oldid=2550752 U3227684] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&oldid=2552580 U3223114] <!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides. ;2022 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&oldid=2420008 U3203545] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&oldid=2419862 U3210431] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&oldid=2420416 U3216256] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&oldid=2419390 Icantchooseone] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&oldid=2420715 U3217287] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&oldid=2418874 Molzaroid] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&oldid=2420232 Ana028] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&oldid=2418246 U3211603] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&oldid=2417630 U3223109] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&oldid=2420956 U3213441] ;2021 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&action=history U3186377] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&action=history U3202904] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&oldid=2313868 U3194166] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&oldid=2313844 U3187741] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&oldid=2310813 U3187874] * [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&oldid=2312677 U3202854] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&oldid=2311466 U3187178] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&oldid=2311388 J.Payten] ;2020 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&oldid=2191350 U3170940] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&oldid=2198005 U3190210] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&oldid=2198079 U3160224] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&oldid=2194797 U3190467] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&oldid=2199869 NUMBLA0371] ;2019 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&oldid=2051716 U3172958] ;2018 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&oldid=1915671 U3160212] ;2017 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&oldid=1730836 U3122707] --> ==Licensing== Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener. ==See also== * [[/Checklist|Topic development — Checklist]] * Marking and feedback ** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]] ** [[Template:METF|Official feedback template]] * Tutorials ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Topic| ]] [[Category:Motivation and emotion guidelines]] 7onh4q05u7tcjbea8n9ha6xmwtr16zf 2816065 2816064 2026-06-17T05:37:54Z Jtneill 10242 /* References (10%) */ 2816065 wikitext text/x-wiki {{title|Topic development — Guidelines}} <div style="text-align: center;">''Chapter plan and user page'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 08 |day = 14 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester -->{{:Motivation and emotion/Assessment/Chapter/Contents}}</div> {{TOCright}} ==Overview== * Weight: 10% * Due: {{/Due}} * Tasks ** Create a Wikiversity user account ** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]] ** Build wiki editing skills by developing a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] which consists of: *** Title and sub-title *** Headings (and possibly sub-headings) **** Overview **** 3-5 other top-level headings **** Conclusion **** See also (with at least 2 lins (1 Wikiveristy and 1 Wikipedia)) **** References (at least 6) **** External links (at least 2) *** Key points for each section (and sub-section) *** Figure (at least 1) *** Learning feature (plan at least 1) ** Create a Wikiversity user page *** Introduce yourself *** Summarise at least three different types of social contributions on your Wikiversity user page * Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]] * Guidance for this assignment is provided in Module 1: ** [[Motivation and emotion/Lectures/Introduction|Lecture 01]] ** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] *Feedback will be provided to help guide drafting of the full [[Motivation and emotion/Assessment/Chapter|book chapter]] *Marks and feedback should be returned before Census Date (end of Week 4) **Marks will be available via {{Motivation and emotion/Canvas}} **Written feedback will be available via the topic's Wikiversity discussion page *Follow up if you don't understand or would like more feedback ==Extensions and late submissions== * Extension requests require an Extension Application Form to be submitted via {{Motivation and emotion/Canvas}} with appropriate documentary evidence * Submissions are accepted up to 3 days late (-10% per day late) * If you don't submit this assessment on time, withdrawal from the unit before Census Date (end of Week 4) is recommended ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic. |- style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Be professional — communicate effectively | Communicate your ideas by sharing a chapter plan; provide feedback on other plans. |- style="vertical-align:top;" | Be professional — display initiative and drive, and use organisation skills to plan and manage workload | Get organised by selecting a topic and submitting an on-time chapter plan. |- style="vertical-align:top;" | Be a lifelong learner — evaluate and adopt new technology | Learn how to edit in a collaborative, online environment. |} ==Instructions== Follow these instructions for the topic development: * Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of: *# Title and sub-title (pre-approved or negotiated) *# Overview *# 3-5 other top-level headings *# Key points for each heading/sub-heading with citations *# 1+ relevant figure(s) *# 1+ actual or planned learning feature *# 6+ references *# 4+ resources *#* See also: 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article) *#* External links: 2+ external links (to external resources) *# Wikiversity user page *#* self-introduction which links to the chapter being worked on *#* Social contributions in a numbered list with a summary and direct link to evidence *#** 1 direct edit to improve another book chapter (past or present) *#** 1 talk page comment on another book chapter (past or present) *#** 1 {{Motivation and emotion/Canvas}} discussion post * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * <span id="Word count">Length (Word count):</span> There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700). * Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments ==Template== {{:Motivation and emotion/Assessment/Topic/Quickstarttip}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] {{anchor|Title}} ===Title and sub-title (10%)=== * Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}}) * Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] * Do not include user name; authorship is as per the page's editing history {{anchor|Headings}} ===Headings (10%)=== * Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links) * Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading. * The top-level headings should align with the sub-title and focus questions * Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]]) {{anchor|Overview}} ===Overview (10%)=== * A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] * At least 3 bullet points outlining the "problem" (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]] * 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] {{anchor|Key points}} ===Key points (10%)=== * At least 3 bullet points per section (i.e., per heading or sub-heading) * Overview the most relevant theory(ies), including key citations * Overview the most relevant research, including key citations * Provide at least 1 introductory bullet point before branching into sub-sections * Address the problem (i.e., answer the question in the sub-title) {{Anchor|Figure}} ===Figure (10%)=== * Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]]. * Number each figure sequentially (e.g., Figure 1, Figure 2 etc.) * Include a descriptive caption that connects the figure to the text * Cite each figure at least once in the main text (e.g., see Figure 1) * Optimise image display size to make it easy to read (i.e., not too big or too small) {{Anchor|Learning feature}} ===Learning feature (10%)=== * In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,: ** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] ** Internal (wiki) links: *** At least 1 embedded link to a relevant book chapter *** At least 1 embedded link to a relevant Wikipedia article * Quiz question with correct and incorrect answers ** Table with an APA style caption {{anchor|References}} ===References (10%)=== * Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]]) * Each source should be cited at least once in the key points * Include a balance of key theoretical and key research articles {{anchor|Resources}} ===Resources (10%)=== * '''See also''' (heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article) ** Provide at least 1 bullet-pointed: *** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter *** internal wiki link to a relevant Wikipedia page ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., Book chapter, 2023) ** Use alphabetical order * '''External links''' (heading): Provide at least 2 external links to key internet resources ** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles) ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., The Conversation) ** Use alphabetical order {{anchor|User page}} ===User page (10%)=== * Create a Wikiversity user page for your user account * Edit the user page to provide information about yourself * Recommended headings: ** About me ** Book chapter I'm working on *** Include an internal (wiki) link to the chapter page ** Social contributions * Consider linking to your other online profiles {{anchor|Social contribution}} {{anchor|Socialcontribution}} ===Social contribution (10%)=== * On your Wikiversity user page, summarise and link to direct evidence that you have made at least 3 different types of contributions: ** direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics) ** provide useful feedback by commenting on a book chapter's talk page (current or previous topic talk pages) ** post to the {{Motivation and emotion/Canvas}} discussion forum<!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --> * [[Motivation and emotion/Wikiversity/Social contributions|More info]] ==Grade descriptions== This section describes typical characteristics of topic developments at each grade level, based on the [[#Marking criteria|marking criteria]]. {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | A clear, complete, easy to understand plan is presented. Considerable depth and breadth of theoretical and research knowledge of the topic is demonstrated via the scope and detail within the plan. All recommended sections are provided. The development of the plan illustrates that the author has actively engaged in developing skills required for collaborative online writing and editing (e.g., interwiki links are provided for key terms, responses are made to comments on the chapter talk page). There are citations to more than 6 key academic sources with references provided in APA style. The author introduces themself on their Wikiversity user page and summarises and provides directly verifiable evidence of editing another chapter, comment provided on another chapter's talk page, and posting to the discussion forum. |- | style="vertical-align:top;" | '''DI (Distinction)''' | A very good, understandable plan is presented. The plan includes key relevant theory and research, with relevant references. The material is well organised into sections, with minimal spelling and grammar issues. There is good evidence that the author has developed the capacity to work effectively in the collaborative editing environment. The author's user page is set up and links to evidence of social contributions. However, there is at least 1 area for improvement. |- | style="vertical-align:top;" | '''CR (Credit)''' | A competent plan is presented. The plan includes the main ideas and sections necessary for developing a good chapter about the topic. Some aspects of the plan, however, may be missing, limited, or problematic. For example, the headings and structure may be under-developed, the reference list may indicate a lack of depth in investigation of the topic, use of wiki links and/or images could often be improved, and/or user page set-up feedback about other chapters may not have been completed. |- | style="vertical-align:top;" | '''P (Pass)''' | A basic, sufficient plan is presented, however there may be incomplete coverage of relevant theory and research, and/or a lack of depth or breadth in conceptualising the chapter. The chapter plan covers basic theory and research about the topic, but lacks detail about how the concepts will be brought together to help address the topic. A basic heading structure is presented, but is likely to need more sections and/or improved formatting or organisation. Spelling and grammar problems are often evident. Citation and referencing tends to be missing or limited in scope and quality (e.g., top peer-reviewed citations about the topic haven't been cited). These plans usually have very brief edit histories (e.g., less than 24 hours) and are often noticeably shorter than plans which attract higher grades. Authors often haven't set up an informative user page or provided evidence of engagement with the development of other chapter plans. |- | style="vertical-align:top;" | '''F (Fail)''' | The plan is insufficient and/or incomplete. Major gaps and/or errors in content are evident. Little evidence of awareness of relevant theory, research, and use of peer-reviewed references. These plans typically have under-developed heading structures and do not illustrate the use of key editing skills. Written expression is often undermined by poor spelling and/or grammar. These plans typically have very brief editing histories (e.g., consist of a few, last minute edits). There is generally no evidence of active engagement with the development of other chapters. |} ==Examples== ;About * Below are some examples of topic development submissions which received 100% * The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions * It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter) ;2025 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&oldid=2726043 Elina.jean.r] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&oldid=2727975 U3254978] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&oldid=2739659 U3237996] ;2024 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&oldid=2644098 U3216883] ;2023 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&oldid=2550752 U3227684] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&oldid=2552580 U3223114] <!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides. ;2022 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&oldid=2420008 U3203545] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&oldid=2419862 U3210431] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&oldid=2420416 U3216256] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&oldid=2419390 Icantchooseone] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&oldid=2420715 U3217287] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&oldid=2418874 Molzaroid] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&oldid=2420232 Ana028] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&oldid=2418246 U3211603] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&oldid=2417630 U3223109] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&oldid=2420956 U3213441] ;2021 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&action=history U3186377] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&action=history U3202904] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&oldid=2313868 U3194166] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&oldid=2313844 U3187741] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&oldid=2310813 U3187874] * [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&oldid=2312677 U3202854] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&oldid=2311466 U3187178] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&oldid=2311388 J.Payten] ;2020 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&oldid=2191350 U3170940] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&oldid=2198005 U3190210] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&oldid=2198079 U3160224] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&oldid=2194797 U3190467] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&oldid=2199869 NUMBLA0371] ;2019 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&oldid=2051716 U3172958] ;2018 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&oldid=1915671 U3160212] ;2017 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&oldid=1730836 U3122707] --> ==Licensing== Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener. ==See also== * [[/Checklist|Topic development — Checklist]] * Marking and feedback ** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]] ** [[Template:METF|Official feedback template]] * Tutorials ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Topic| ]] [[Category:Motivation and emotion guidelines]] ru76553d2qjndmt8nrxpv0yy4e63ctq 2816066 2816065 2026-06-17T05:38:27Z Jtneill 10242 /* See also */ 2816066 wikitext text/x-wiki {{title|Topic development — Guidelines}} <div style="text-align: center;">''Chapter plan and user page'' <!-- ---------------------------------- ---> <!-- Count down --> <!-- ---------------------------------- ---><!-- {{countdown |year = 2025 |month = 08 |day = 14 |hour = 23 |minute = 0 |second = 0 |event = this assessment is due }} --> <!-- {{Motivation and emotion/Assessment/In development}} --> <!-- Show this during semester -->{{:Motivation and emotion/Assessment/Chapter/Contents}}</div> {{TOCright}} ==Overview== * Weight: 10% * Due: {{/Due}} * Tasks ** Create a Wikiversity user account ** Select or negotiate an approved topic in the [[Motivation and emotion/Book/2026|2026 table of contents]] ** Build wiki editing skills by developing a plan for the [[Motivation and emotion/Assessment/Chapter|book chapter]] which consists of: *** Title and sub-title *** Headings (and possibly sub-headings) **** Overview **** 3-5 other top-level headings **** Conclusion **** See also (with at least 2 lins (1 Wikiveristy and 1 Wikipedia)) **** References (at least 6) **** External links (at least 2) *** Key points for each section (and sub-section) *** Figure (at least 1) *** Learning feature (plan at least 1) ** Create a Wikiversity user page *** Introduce yourself *** Summarise at least three different types of social contributions on your Wikiversity user page * Follow the detailed [[#Instructions|instructions]] and address the [[#Marking criteria|marking criteria]] * Guidance for this assignment is provided in Module 1: ** [[Motivation and emotion/Lectures/Introduction|Lecture 01]] ** [[Motivation and emotion/Lectures/Historical development and assessment skills|Lecture 02]] ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ==Marking and feedback== *Submissions will be marked according to the [[#Marking criteria|marking criteria]] *Feedback will be provided to help guide drafting of the full [[Motivation and emotion/Assessment/Chapter|book chapter]] *Marks and feedback should be returned before Census Date (end of Week 4) **Marks will be available via {{Motivation and emotion/Canvas}} **Written feedback will be available via the topic's Wikiversity discussion page *Follow up if you don't understand or would like more feedback ==Extensions and late submissions== * Extension requests require an Extension Application Form to be submitted via {{Motivation and emotion/Canvas}} with appropriate documentary evidence * Submissions are accepted up to 3 days late (-10% per day late) * If you don't submit this assessment on time, withdrawal from the unit before Census Date (end of Week 4) is recommended ==Learning outcomes== How the unit's [[Motivation and emotion/About/Learning outcomes|learning outcomes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Learning outcome''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Integrate theories and current research towards explaining the role of motivation and emotions in human behaviour. | Identify the main psychological theories and peer-reviewed research which can be used to explain a specific motivation or emotion topic. |- style="vertical-align:top;" | Critically apply knowledge of motivation or emotion to an indepth understanding of a specific topic in this field. | Propose how psychological knowledge can be applied to a specific topic to improve motivational and emotional lives. |} ==Graduate attributes== How the unit's [[Motivation and emotion/About/Graduate attributes|graduate attributes]] are addressed by this assessment exercise: {| border=1 cellpadding=5 cellspacing="0" background:transparent style="width:90%; margin: auto;" |- style="vertical-align:top;" | style="width:40%;" | '''Graduate attribute''' | style="width:60%;" | '''Assessment task''' |- style="vertical-align:top;" | Be professional — communicate effectively | Communicate your ideas by sharing a chapter plan; provide feedback on other plans. |- style="vertical-align:top;" | Be professional — display initiative and drive, and use organisation skills to plan and manage workload | Get organised by selecting a topic and submitting an on-time chapter plan. |- style="vertical-align:top;" | Be a lifelong learner — evaluate and adopt new technology | Learn how to edit in a collaborative, online environment. |} ==Instructions== Follow these instructions for the topic development: * Develop a plan for a [[Motivation and emotion/Assessment/Chapter|chapter]] which consists of: *# Title and sub-title (pre-approved or negotiated) *# Overview *# 3-5 other top-level headings *# Key points for each heading/sub-heading with citations *# 1+ relevant figure(s) *# 1+ actual or planned learning feature *# 6+ references *# 4+ resources *#* See also: 2+ internal links (1 to Wikiversity (e.g., another book chapter) and 1 to a Wikipedia article) *#* External links: 2+ external links (to external resources) *# Wikiversity user page *#* self-introduction which links to the chapter being worked on *#* Social contributions in a numbered list with a summary and direct link to evidence *#** 1 direct edit to improve another book chapter (past or present) *#** 1 talk page comment on another book chapter (past or present) *#** 1 {{Motivation and emotion/Canvas}} discussion post * [[Motivation and emotion/Assessment/Using generative AI|Generative AI]] may be used with appropriate acknowledgement * <span id="Word count">Length (Word count):</span> There is no minimum or maximum length. Top-ranked topic development [[#Examples|examples]] range from 875 to 2900 words (average 1700). * Submit a PDF of the topic development via {{Motivation and emotion/Canvas}}, with the title, sub-title, and user name in the submission comments ==Template== {{:Motivation and emotion/Assessment/Topic/Quickstarttip}} ==Marking criteria== [[File:Balanced scales.svg|right|125px]] {{anchor|Title}} ===Title and sub-title (10%)=== * Use the approved wording, [[w:Letter case#Sentence case|casing]], etc. for the title and sub-title (i.e., as per the {{Motivation and emotion/Book}}) * Do not include additional bold, italics, or change font size from the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] * Do not include user name; authorship is as per the page's editing history {{anchor|Headings}} ===Headings (10%)=== * Use the standard headings recommended in the [[Template:Motivation_and_emotion/Book_chapter_structure|book chapter template]] (i.e., Overview, Conclusion, References, See also, External links) * Provide 3 to 6 informative top-level headings between the Overview and Conclusion. These sections may each contain 2 to 5 sub-headings; avoid sections with only 1 sub-heading. * The top-level headings should align with the sub-title and focus questions * Headings should use [[w:Letter case#Sentence case|sentence casing]] (see also [[:Template:Heading casing|heading casing]]) {{anchor|Overview}} ===Overview (10%)=== * A scenario or case study (real or fictional), in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] * At least 3 bullet points outlining the "problem" (i.e., explain the key concept(s) and importance of the topic)—to be expanded into sentences and paragraphs for the [[Motivation and emotion/Assessment/Chapter|book chapter]] * 3 to 5 [[Motivation and emotion/Assessment/Chapter/Focus questions|focus questions]] that unpack the topic and address the sub-title, in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] {{anchor|Key points}} ===Key points (10%)=== * At least 3 bullet points per section (i.e., per heading or sub-heading) * Overview the most relevant theory(ies), including key citations * Overview the most relevant research, including key citations * Provide at least 1 introductory bullet point before branching into sub-sections * Address the problem (i.e., answer the question in the sub-title) {{Anchor|Figure}} ===Figure (10%)=== * Display at least 1 relevant figure. See [[Template:Motivation and emotion/Book chapter structure#Figures|example]]. * Number each figure sequentially (e.g., Figure 1, Figure 2 etc.) * Include a descriptive caption that connects the figure to the text * Cite each figure at least once in the main text (e.g., see Figure 1) * Optimise image display size to make it easy to read (i.e., not too big or too small) {{Anchor|Learning feature}} ===Learning feature (10%)=== * In addition to the scenario in the Overview, include at least 1 of the following learning features e.g.,: ** Another scenario/case study: A follow-up or second scenario/case study in the main body in a [[Motivation and emotion/Wikiversity/Feature box|feature box]] ** Internal (wiki) links: *** At least 1 embedded link to a relevant book chapter *** At least 1 embedded link to a relevant Wikipedia article * Quiz question with correct and incorrect answers ** Table with an APA style caption {{anchor|References}} ===References (10%)=== * Provide at least 6 APA style references to the best peer-reviewed sources about the topic (e.g., see [[Motivation and emotion/Journals|list of motivation and emotion journals]]) * Each source should be cited at least once in the key points * Include a balance of key theoretical and key research articles {{anchor|Resources}} ===Resources (10%)=== * '''See also''' (heading): Provide at least 2 internal (wiki) links (1 to a Wikiversity article; 1 to a Wikipedia article) ** Provide at least 1 bullet-pointed: *** [[Help:Contents/Links#Interwiki_links|internal (wiki) link]] to a relevant book chapter *** internal wiki link to a relevant Wikipedia page ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., Book chapter, 2023) ** Use alphabetical order * '''External links''' (heading): Provide at least 2 external links to key internet resources ** Provide at least 2 bullet-pointed [[Help:Contents/Links#External_links|external link]]s to key internet resources (not Wikiversity or Wikipedia or academic articles) ** The linked text is the same as the name of the target page using [[w:Letter case#Sentence casing|sentence casing]] ** Include the source in parentheses after the link (e.g., The Conversation) ** Use alphabetical order {{anchor|User page}} ===User page (10%)=== * Create a Wikiversity user page for your user account * Edit the user page to provide information about yourself * Recommended headings: ** About me ** Book chapter I'm working on *** Include an internal (wiki) link to the chapter page ** Social contributions * Consider linking to your other online profiles {{anchor|Social contribution}} {{anchor|Socialcontribution}} ===Social contribution (10%)=== * On your Wikiversity user page, summarise and link to direct evidence that you have made at least 3 different types of contributions: ** direct edit to improve a [[Motivation and emotion/Book|book chapter page]] (current or previous topics) ** provide useful feedback by commenting on a book chapter's talk page (current or previous topic talk pages) ** post to the {{Motivation and emotion/Canvas}} discussion forum<!-- or contribute to the {{Motivation and emotion/Hashtag}} X hashtag --> * [[Motivation and emotion/Wikiversity/Social contributions|More info]] ==Grade descriptions== This section describes typical characteristics of topic developments at each grade level, based on the [[#Marking criteria|marking criteria]]. {| border=1 cellpadding=7 cellspacing=0 style = "background:transparent; width:90%" ! Grade ! Description |- | style="width:140px; vertical-align:top;" | '''HD (High Distinction)''' | A clear, complete, easy to understand plan is presented. Considerable depth and breadth of theoretical and research knowledge of the topic is demonstrated via the scope and detail within the plan. All recommended sections are provided. The development of the plan illustrates that the author has actively engaged in developing skills required for collaborative online writing and editing (e.g., interwiki links are provided for key terms, responses are made to comments on the chapter talk page). There are citations to more than 6 key academic sources with references provided in APA style. The author introduces themself on their Wikiversity user page and summarises and provides directly verifiable evidence of editing another chapter, comment provided on another chapter's talk page, and posting to the discussion forum. |- | style="vertical-align:top;" | '''DI (Distinction)''' | A very good, understandable plan is presented. The plan includes key relevant theory and research, with relevant references. The material is well organised into sections, with minimal spelling and grammar issues. There is good evidence that the author has developed the capacity to work effectively in the collaborative editing environment. The author's user page is set up and links to evidence of social contributions. However, there is at least 1 area for improvement. |- | style="vertical-align:top;" | '''CR (Credit)''' | A competent plan is presented. The plan includes the main ideas and sections necessary for developing a good chapter about the topic. Some aspects of the plan, however, may be missing, limited, or problematic. For example, the headings and structure may be under-developed, the reference list may indicate a lack of depth in investigation of the topic, use of wiki links and/or images could often be improved, and/or user page set-up feedback about other chapters may not have been completed. |- | style="vertical-align:top;" | '''P (Pass)''' | A basic, sufficient plan is presented, however there may be incomplete coverage of relevant theory and research, and/or a lack of depth or breadth in conceptualising the chapter. The chapter plan covers basic theory and research about the topic, but lacks detail about how the concepts will be brought together to help address the topic. A basic heading structure is presented, but is likely to need more sections and/or improved formatting or organisation. Spelling and grammar problems are often evident. Citation and referencing tends to be missing or limited in scope and quality (e.g., top peer-reviewed citations about the topic haven't been cited). These plans usually have very brief edit histories (e.g., less than 24 hours) and are often noticeably shorter than plans which attract higher grades. Authors often haven't set up an informative user page or provided evidence of engagement with the development of other chapter plans. |- | style="vertical-align:top;" | '''F (Fail)''' | The plan is insufficient and/or incomplete. Major gaps and/or errors in content are evident. Little evidence of awareness of relevant theory, research, and use of peer-reviewed references. These plans typically have under-developed heading structures and do not illustrate the use of key editing skills. Written expression is often undermined by poor spelling and/or grammar. These plans typically have very brief editing histories (e.g., consist of a few, last minute edits). There is generally no evidence of active engagement with the development of other chapters. |} ==Examples== ;About * Below are some examples of topic development submissions which received 100% * The links go to snapshots of pages as submitted for the topic development; these are not the final book chapter submissions * It is possible to get full marks using only bullet points, however some examples below go beyond the requirements for 100% (e.g., involve drafting a full chapter) ;2025 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2729232 Metacognition and emotional regulation] - [https://en.wikiversity.org/w/index.php?title=User:Elina.jean.r&oldid=2726043 Elina.jean.r] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Motivation_for_using_AI_companions&oldid=2728874 Motivation for using AI companions] - [https://en.wikiversity.org/w/index.php?title=User:U3254978&oldid=2727975 U3254978] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Self-determination_theory_and_social_media_use&oldid=2740305 Self-determination theory and social media use] - [https://en.wikiversity.org/w/index.php?title=User:U3237996&oldid=2739659 U3237996] ;2024 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2024/Groups_and_individual_motivation_reduction&oldid=2644110 Groups and individual motivation reduction] - [https://en.wikiversity.org/w/index.php?title=User:U3216883&oldid=2644098 U3216883] ;2023 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Bedtime_procrastination&oldid=2550954 Bedtime procrastination] - [https://en.wikiversity.org/w/index.php?title=User:U3227684&oldid=2550752 U3227684] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2023/Conspiracy_theory_motivation&oldid=2551397 Conspiracy theory motivation] - [https://en.wikiversity.org/w/index.php?title=User:U3223114&oldid=2552580 U3223114] <!-- * The topic development requirements and weighting increased in 2023 from 5% to 10%. So, the examples from 2022 and earlier may not warrant full marks if assessed against the 2023-present criteria. They should nevertheless serve as useful guides. ;2022 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Compassion&oldid=2420004 Compassion] — [https://en.wikiversity.org/w/index.php?title=User:U3203545&oldid=2420008 U3203545] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Childhood_trauma_and_subsequent_drug_use&oldid=2429214 Childhood trauma and subsequent drug use] — [https://en.wikiversity.org/w/index.php?title=User:U3210431&oldid=2419862 U3210431] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Disappointment&oldid=2420355 Disappointment] — [https://en.wikiversity.org/w/index.php?title=User:U3216256&oldid=2420416 U3216256] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Fear&oldid=2419996 Fear] — [https://en.wikiversity.org/w/index.php?title=User:Icantchooseone&oldid=2419390 Icantchooseone] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Financial_investing,_motivation,_and_emotion&oldid=2420729 Financial investing, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3217287&oldid=2420715 U3217287] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Money_priming,_motivation,_and_emotion&oldid=2420693 Money priming, motivation, and emotion] — [https://en.wikiversity.org/w/index.php?title=User:Molzaroid&oldid=2418874 Molzaroid] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Nature_therapy&oldid=2420231 Nature therapy] — [https://en.wikiversity.org/w/index.php?title=User:Ana028&oldid=2420232 Ana028] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Video_conferencing_fatigue&oldid=2421389 Video conferencing fatigue] - [https://en.wikiversity.org/w/index.php?title=User:U3211603&oldid=2418246 U3211603] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Window_of_tolerance&oldid=2419756 Window of tolerance] — [https://en.wikiversity.org/w/index.php?title=User:U3223109&oldid=2417630 U3223109] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Work_and_flow&oldid=2421675 Work and flow] — [https://en.wikiversity.org/w/index.php?title=User:U3213441&oldid=2420956 U3213441] ;2021 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Affective_disorders&oldid=2314003 Affective disorders] — [https://en.wikiversity.org/w/index.php?title=User:U3186377&action=history U3186377] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Cognitive_dissonance_and_motivation&oldid=2313463 Cognitive dissonance and motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3202904&action=history U3202904] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Domestic_violence_motivation&oldid=2313842 Domestic violence motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3194166&oldid=2313868 U3194166] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Fantasy_and_sexual_motivation&oldid=2313839 Fantasy and sexual motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3187741&oldid=2313844 U3187741] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Laziness&oldid=2312068 Laziness] — [https://en.wikiversity.org/w/index.php?title=User:U3187874&oldid=2310813 U3187874] * [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2021/Non-English_emotion_words Non-English emotion words] — [https://en.wikiversity.org/w/index.php?title=User:U3202854&oldid=2312677 U3202854] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Positive_illusions_about_the_self&oldid=2312873 Positive illusions about the self] — [https://en.wikiversity.org/w/index.php?title=User:U3187178&oldid=2311466 U3187178] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2021/Torture_motivation&oldid=2311842 Torture motivation] — [https://en.wikiversity.org/w/index.php?title=User:J.Payten&oldid=2311388 J.Payten] ;2020 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Body_image_flexibility&oldid=2196896 Body image flexibility] — [https://en.wikiversity.org/w/index.php?title=User:U3170940&oldid=2191350 U3170940] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Emotional_self-efficacy&oldid=2200012 Emotional self-efficacy] — [https://en.wikiversity.org/w/index.php?title=User:U3190210&oldid=2198005 U3190210] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Guilty_pleasure&oldid=2196391 Guilty pleasure] — [https://en.wikiversity.org/w/index.php?title=User:U3160224&oldid=2198079 U3160224] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Meta-emotion&oldid=2199480 Meta-emotion] — [https://en.wikiversity.org/w/index.php?title=User:U3190467&oldid=2194797 U3190467] * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2020/Methamphetamine_and_emotion&oldid=2199878 Methamphetamine and emotion] — [https://en.wikiversity.org/w/index.php?title=User:NUMBLA0371&oldid=2199869 NUMBLA0371] ;2019 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2019/Growth_mindset_development&oldid=2052186 Growth mindset development] — [https://en.wikiversity.org/w/index.php?title=User:U3172958&oldid=2051716 U3172958] ;2018 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2018/Familicide_motivation&oldid=1916838 Familicide motivation] — [https://en.wikiversity.org/w/index.php?title=User:U3160212&oldid=1915671 U3160212] ;2017 * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2017/Awe_and_well-being&oldid=1730944 Awe and well-being] — [https://en.wikiversity.org/w/index.php?title=User:U3122707&oldid=1730836 U3122707] --> ==Licensing== Contributions to Wikiversity are made under [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons 4.0 ShareAlike] (CC-BY-SA 4.0) and [http://www.gnu.org/copyleft/fdl.html GFDL] licenses. These licenses give permission for others to edit and re-use contributed content, with appropriate acknowledgement. These licenses are irrevocable.For more information, see the [[wmf:Terms of use|Wikimedia Foundation's Terms of use]]. If you do not wish to contribute your work under these licenses, discuss [[Motivation and emotion/Assessment/Alternative|alternative assessment]] options with the unit convener. ==See also== * [[/Checklist|Topic development — Checklist]] * Marking and feedback ** [[Motivation and emotion/Assessment/Topic/Feedback|General feedback]] ** [[Template:METF|Feedback template]] * Tutorials ** [[Motivation and emotion/Tutorials/Topic selection|Tutorial 01: Topic selection]] ** [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02: Wiki editing]] * [[Motivation and emotion/Assessment/Using generative AI|Using generative AI]] {{Motivation and emotion/Assessment/Navigation}} [[Category:Motivation and emotion/Assessment/Topic| ]] [[Category:Motivation and emotion guidelines]] 5ry8qm4l7w0gwkafwiedw9cz3xij5b7 0 225954 2816025 2544143 2026-06-16T21:58:06Z Jtneill 10242 removed [[Category:Motivation and emotion/Book/Therapy]] using [[Help:Gadget-HotCat|HotCat]] 2816025 wikitext text/x-wiki {{title|Rational emotive behaviour therapy:<br>How can REBT help to change our emotions?}} {{MECR3|https://youtu.be/yBehqxLDqY4}} __TOC__ ==Overview== Developed by [[wikipedia:Albert_Ellis|Albert Ellis]] in the mid-1950s, Rational Emotive Behaviour Therapy (REBT) was the first of the [[wikipedia:Cognitive_behavioral_therapy|cognitive behaviour therapies]] (CBT) and continues to be one of the major CBT approaches (Bond & Dryden, 2000). After practicing psychoanalysis with [[wikipedia:Sigmund_Freud|Sigmund Freud]], Ellis came to see flaws with his approach. Many of the answers he was looking for came from philosophy, an interest from his youth. He found that contemporary theories complimented{{sp}}, and worked well with, theories from philosophers from the past (Ellis & Joffe Ellis, 2011). Anxiety, panic, depression, and fear observed as avoidance, are all [[wikipedia:Emotion|emotions]] often requiring professional therapy{{rewrite}}. In dealing with these emotions, REBT teaches that, although almost everyone wants happiness, we readily counteract efforts to reach our goals with our own thoughts and perceptions. During the process of REBT clients learn about the interactional relationship between thoughts, behaviour, and emotions. To prevent detrimental emotional effects, clients learn new ways to view and respond to life's challenges. REBT therapy involves many different techniques and may also include homework assignments. With hard work and practice, REBT promises long-term emotional change. {{robelbox|theme=9|title=Key Topic Questions|icon=Nuvola_apps_kwrite.png|iconwidth=48px}} <div style="{{Robelbox/pad}}"> * Name some of the historical figures that had influence in the creation of REBT? * What is ABC theory? * What type of emotional disturbances can be helped by REBT? * What is the difference between a rational negative thought and an irrational negative thought? * How does REBT help change emotions? </div> {{Robelbox/close}} ==Foundational theories and principles== Some of the historical figures that provided inspiration to Ellis when developing REBT include: [[wikipedia:Gautama_Buddha|Buddha]] - The Buddha used a similar active-directive style with his followers in order to help them learn to question their thoughts and emotions. When their current thinking was not effective, they were asked to consider opening up to different ways of thinking and work hard to change these habits. The Buddhist perspective views life to be rooted in suffering and this suffering is caused by cravings or needs. It is believed that relief from suffering can only be gained by ending these urges and accepting what is apparent. This can be compared to REBT teaching the recognition of irrational beliefs in order to relieve emotional distress (Christopher, 2003). [[wikipedia:Epictetus|Epictetus]] - Epictetus' statement "Men are not disturbed by things but by the view they take of them" (Ellis & Joffe Ellis, 2011, p. 27) is often quoted by Ellis. The power of thought over feelings or emotions, central to REBT, can be seen in this statement. Epictetus was a Roman Stoic philosopher. His accepting and unangry philosophies are often used as part of REBT therapy (Ellis & Joffe Ellis, 2011). Other historical figures noted by Ellis to have influenced the creation of REBT include [[wikipedia:Marcus_Aurelius|Marcus Aurelius]], [[wikipedia:Confucius|Confucius]], [[wikipedia:Laozi|Lao Tzu]], [[wikipedia:Socrates|Socrates]], [[wikipedia:Epicurus|Epicurus]], [[wikipedia:Seneca_the_Elder|Seneca]] and [[wikipedia:Ralph_Waldo_Emerson|Emerson]] (Ellis & Joffe Ellis, 2011) REBT also has influences from many contemporary perspectives including the psychotherapist [[wikipedia:Alfred_Adler|Alfred Adler]] and [[wikipedia:Sigmund_Freud|Freud]]. Constructivism is an idea introduced by [[wikipedia:George_Kelly_(psychologist)|George Kelly]] with his personal construct theory in 1955. Constructivism claims that a person has the ability to construct their own reality by controlling their thoughts, feelings. and actions (McWilliams, 2016). This suggests that emotional difficulties can be caused by one's own thoughts, an idea central to REBT. Ellis had a great deal of interest in the work of [[wikipedia:John_B._Watson|John B Watson]] who founded [[wikipedia:Behaviorism|Behaviourism]]. Although Watson rejected the idea of thoughts and introspection being central to emotional difficulties, he provided Ellis with information and understandings contributing to REBT. Ellis found great value in Watson's insistence on an empirical approach to achieving definitive outcomes (Carpintero, 2004). Over the years of gathering this information Ellis learned that people have more control of their emotions than they may realise. REBT holds that people are the main contributors to their own emotional distress due to their thoughts. REBT claims that people can counteract emotional distress by learning to observe their thoughts and working on changing problematic thinking (Ellis, 1990). [[File:ABC Image II.svg|thumb|''Figure 2''. People mistakenly believe the event causes the consequence|300x300px|left]] [[File:ABC Image III.svg|thumb|''Figure 3''. REBT perspective on the influence of a person's thoughts in relation to the consequence after the event|300x300px]] ===ABC theory=== Life provides us with a variety of experiences, some we look forward to and enjoy, some are neutral and others are perceived as activating events (A). It is these events which are often viewed as the cause of a person's emotional disturbance. REBT holds that it is not the situation or event causing emotional problems. It asserts that people mainly cause the difficulties they experience with self-disturbing beliefs (B) about the event, resulting in the difficult emotional consequence (C). Therefore, contrary to traditional understanding where A determines C, the combination of both A and B determines the resulting emotional distress at C. ===Rational and irrational thoughts=== It is claimed that if a difficult situation should arise, a person thinking rationally will experience healthy negative emotions like sadness, remorse and disappointment. However, a person thinking in irrational ways will experience unhealthy negative emotions like depression, guilt and shame. When As are viewed negatively and perceived in a rational manner a person will think along these lines, "I don't like As and I wish they didn't happen but they do and I will deal with it". This would enable the person to deal with the event. If, on the other hand, the person was to be irrational about it they may think, "This A is terrible and I can't stand it!" In this circumstance unhealthy negative emotions will likely result. (Ellis & Joffe Ellis, 2011). See Table 1 for further comparison of rational and irrational thoughts. Table 1<br>''Comparison of Rational and Irrational Thoughts as Proposed by REBT'' {| class="wikitable" !Rational thoughts !Irrational thoughts |- |Founded in science<br>Assessed with good reason<br>Claim a preference for things to be the way the person wants, not demanding it<br>Is forgiving of self, others and life<br>Has a high level of tolerance and acceptance<br>Results in appropriate and healthy emotions (Dryden, 2010). |Uses words like should, must and ought<br>When things go wrong it is a catastrophe, awful or exaggerated<br>Is demanding<br>Is judgemental<br>Has a low tolerance of frustration<br>Emotional results are unhealthy and debilitating (Dryden, 2010). |} REBT theory states that irrational beliefs fall into four main categories: # Low frustration tolerance # Demandingness # Global self-downing/evaluation # Awfulising/catastrophising (Szentagotai, Lupu, & Cosman, 2008) ===Interconnectedness of thoughts, emotions and behaviour=== [[File:Thoughts, feelings and behaviour.svg|thumb| ''Figure 4''. REBT proposes thoughts, emotions and behaviours all interact and influence each other |300x300px|left]] Another principle of REBT is the interconnectedness of our thoughts, emotions and behaviour. It is believed that thoughts, emotions, and behaviour are all interrelated contributions to both B (beliefs) and C (consequence), thus determining a person's healthy or unhealthy emotional experience. ===''Must''urbatory beliefs and the three basic musts=== Musts, shoulds, and oughts are often part of irrational thinking. REBT hypothesises that people frequently learn to become ''must''urbators from significant others like parents, teachers, friends, and other highly regarded social acquaintances. Perhaps even more problematic is the innate tendency that people have to want more than they have, increasing that want to ''must'' have. REBT aims to prevent clients using three basic musts as it is believed they result in emotional difficulties. Use of the word ''must'' is to be avoided in contexts such as "I must always perform well or else I am no good", "others must treat me well or they are no good" and "the world or life must always treat me well or life is no good". Each of these uses of the word must will contribute to specific emotional problems. "I must always, perform well or else I am no good" is believed to result in anxiety, depression, worthlessness, despair, shame and guilt. "Others must treat me well or they are no good" is thought to cause anger, vindictiveness, rage, passive aggression, and violence. "The world or life must always treat me well or life is no good" will leave the person experiencing frustration, intolerance, self-pity, procrastination, and depression (Ellis, 1999). ===Meta-emotional problems (emotional disturbance about emotional disturbance)=== When a person has an uncomfortable emotional response to an activating event this is considered the primary experience. Commonly referred to as secondary emotional problems, meta-emotional problems refer to detrimental psychological behaviour in response to the primary experience. This experience is a magnification of the primary experience due to the focus on, and response to, the primary emotion. This causes the person to experience the negative emotion in an even more intense manner. Some examples are experiencing anxiety about being anxious, being depressed about being depressed, putting yourself down about putting yourself down (Dryden & Neenan, 2004). ==Techniques== REBT is multi-modal, and as such, has integrated aspects from cognitive, emotive, and behavioural therapies. One of the main aims is for clients to gain unconditional self-acceptance and to become self-sufficient, not relying on the psychotherapist and needing to return to therapy for the same problem. It also recommends that a person live with humour and a healthy perspective. A person taking themselves and others too seriously is seen to lead to detrimental thinking causing emotional discomfort. As such, REBT uses a variety of techniques and often these are aimed at lightening the mood, including humorous songs, and joking around during a session (Ellis & Joffe Ellis, 2011). ===Unconditional acceptance=== [[File:Sunset-day-summer-sky-90762.jpg|thumb|''add figure caption here'']] Due to the detrimental effects of ''must''urbatory beliefs, REBT teaches that one of the main contributors to emotional disturbance is a lack of self-acceptance, other-acceptance, and life-acceptance. Clients are taught to consistently aim to unconditionally accept themselves, others, and life, especially during difficult times. Examples of opportunities to practice this outside the clinic include: for the self, when one makes a mistake; for others, when a person does something to you that you consider to be disrespectful; towards life, such as rain on your wedding day. ===ABC(DE) theory=== During therapy clients are taught the detrimental effect of inappropriate beliefs or opinions about the A. When concluding a session clients are often asked to partake in homework exercises related to ABC theory. To solidify their understanding, clients are often asked to write up and label the ABC specifics about events that happen in their lives during the time between sessions. This labelling includes all facets of the ABC with particular focus on the B. Labelling the B involves clients being required to recognise the beliefs to be rational or irrational (Rait, Monsen, & Squires, 2010). One of the most effective techniques of REBT is the D - disputing, of ABC theory. As the REBT psychotherapist is considered to be an expert, it is thought appropriate for them to dispute the irrational beliefs of clients. This is done in a considerate but definite manner so as to not leave clients with any doubt about the need to change their way of thinking. The final aspect of the ABC theory is the E - effective new philosophies. During REBT, clients learn to view aspects of life in different ways to those which have been contributing to their emotional difficulties (Ellis, 1974). ===Cognitive=== ABC theory provides the foundation for some of the most effective psychotherapeutic techniques. Clients are taught to find a person who displays emotional well-being, attitude, and behaviours in line with those perceived as ideal by REBT. They are then to aim to use the model's attitudes and behaviours in their own life. Clients are also taught to research, with the aim of finding other people who have lived, or are living, in ways that emulate REBT principles. This is to expand the positive influence in the client's life. Furthermore, clients are to assess the cost to benefit ratio of certain situations. This is done after clients have been taught the negative impact of their detrimental behaviour and learned more beneficial habits. The loss incurred by using the newly learned skill (if any) is then compared to what may be gained. Clients are also encouraged to read books, listen to recordings, and watch videos based on REBT, other CBTs and other inspiring philosophies. This is to reinforce REBT learning. Further reinforcement is gained with clients being advised to teach others REBT habits. This can be done in circumstances focused on learning specifically or through philosophic discussion. Clients are considered to continue benefiting from REBT when they habituate to the practice of being aware of their thoughts and thinking problems through before acting. It is believed this will prevent emotional disturbance by delaying and perhaps preventing emotions being in control. ===Emotive-evocative=== [[File:Emotns.png|left|thumb|147x147px|''add figure caption here'']] Rational emotive imagery (REI) is used in REBT to change unhealthy emotions to more healthy ones. An example of REI can be found [https://www.youtube.com/watch?v=u8ARowMkWNw here]. REI is used to solidify understanding of one of the fundamental concepts of REBT: that all emotions are legitimate, however, some become overwhelming and certain techniques can be used to rectify problems incurred. REI involves clients vividly picturing themselves in a situation experiencing the extreme emotions. While this is happening, the therapist helps the client continue to be in the situation while changing the emotional experience to a more positive one by picturing a much more favourable negative emotional experience. In order for the response to become habit, the technique is to be practiced for at least 30 days (Ellis & Joffe Ellis, 2011). With a similar aim, clients are also encouraged to strongly use coping statements. What makes this an emotive technique is the strength of emphasis placed on the statement. Clients are to repeat the statements whilst making them with vigor. Some examples of statements include: * I don't like this but I ''can'' deal with it! * I may make a mistake but that ''doesn't ever'' make me a failure! * Catastrophes ''do not'' happen to me, just inconveniences! Role-plays are used to evoke emotions by having the person take part in an experience in which they feel uncomfortable emotions. Other people playing parts in the role-play behave in ways so as to increase this discomfort. Observers watching during the role-play critique the client's performance and with the follow up performance all involved aim to find the "shoulds", "musts" and "oughts" contributing to the client's detrimental negative emotional experience (Ellis & Joffe Ellis, 2011). ===Behavioural=== Shame attacking exercises are both emotive-evocative and behavioural exercises. These exercises aim to expose the effect expectation based "shoulds" and "musts" have on the resulting shame experienced by a person. According to REBT, if a person demands they "should not" or "must not" make a mistake and then they do, shame is caused by them judging the mistake. This judgement flows on to oneself. This translation, bad mistake means bad person, is considered false. The shame attacking exercises involves doing something one would normally avoid due to it being considered shameful. While doing this, the person is told to be aware of their thoughts and emotions, being sure not to be embarrassed or put themselves down. REBT uses some other well-known behavioural techniques such as in vivo desensitisation. This method involves a series of assignments leading to desensitisation. For example, a person with a fear of public speaking would initially be given an assignment to present a one minute speech to a close acquaintance. The next assignment would involve a longer speech. Subsequently, a speech of the same length to more people. The length and time would continue to increase until the person reaches their goal (Ellis & Joffe Ellis, 2011). ==General semantics== REBT holds that a person experiencing emotional difficulties is likely to have been unknowingly inducing these experiences upon themselves after gaining understandings about the world as a child. Parents, teachers, and others held in high regard will have imparted understanding with the intention of assisting the child to have a safe journey throughout life. Used in the way intended, the person would react to difficult situations in a well-balanced manner. It is thought however, that this information is distorted by the person's self-talk. REBT adapted principles from Alfred Korzybski, the founder of [[wikipedia:General_semantics|General Semantics]]. One of the semantics adapted includes teaching clients to cease overgeneralising with the aim of having them stop making all-encompassing judgements about the self or others and instead judge the behaviour. Additionally, Korzybski formulated the concept of secondary emotional problems, such as anxiety about being anxious and being depressed about being depressed (Ellis & Joffe Ellis, 2011). Along with [[wikipedia:Karen_Horney|Karen Horney's]] ideas about the "tyranny of shoulds" (as cited in Kerr, 1984), Korzybski contributed to the notion that catastrophising and awfulising has a strong influence on the development of emotional difficulties. According to REBT, we are naturally inclined to overgeneralise, and the influence of significant others adds to this tendency. As such, this behaviour needs to be corrected to reduce emotional difficulty. REBT teaches awareness of absolutistic thinking (must, should, hate, and horrible) and overgeneralisations as it is believed that this way of thinking can lead to the development of emotional problems when expectations are not met. Fortunately, due to the influence of constructivism, REBT teaches that we do not need to think in this way. Even if we do, this can be brought to our attention and these false premises and conclusions can be changed. REBT therapists use different techniques to do this, including education, disputing, discussion, and reasoning, to help their clients toward a more productive way of thinking. ==Work and practice== To increase the efficacy of therapy and reduce further emotional difficulties, REBT emphasises the need for clients to continue working on what they have learned in therapy. This persistence is required not only by the client but also the therapist. Therapists are to choose the most appropriate technique from the many available and persist with it. When it is believed that the most has been gained, it is suggested they try a different technique with the possibility of gaining more benefit for the client (Rait, Monsen, & Squires, 2010). [[File:Wikibooks_library_icon_social_science.svg|right|230x230px|''add Figure caption here'']] ==Efficacy of REBT affecting emotional change== Although REBT is the first CBT and one of the most popular psychotherapeutic approaches, there has been limited research about its efficacy. Ellis and Joffe Ellis (2011) state various reasons for this. For example, The Albert Ellis institute was set up to train students in therapeutic techniques, not to conduct research. It is also suggested that the expense of good research is too high for the institute to afford. Furthermore, the mix of various cognitive, emotive, and behavioural techniques comprising REBT makes it difficult to research. Moreover, the theory of REBT is said to apply to most emotional disturbances, not one or two specifically. Ellis and Joffe Ellis (2011) contend that, considering many REBT techniques are found within CBT, the substantial amount of research provided focusing on CBT validates REBT. For example, in order to support the clinical hypotheses of REBT, Ellis and Joffe Ellis (2011) provided the results of a comprehensive study of meta-analyses performed in relation to CBT. This meta-analysis conducted by Butler, Chapman, Forman and Beck (2006) covered 16 meta-analyses with a total of 9995 participants in 332 separate studies. The studies covered 16 diagnosed emotional disorders and provided 562 comparisons between CBT and other approaches. The results do provide support for cognitive therapy and CBT although it is suggested that future analyses need to provide more investigation into the benefit of CBT in relation to a more diverse population, specific disorders, long-term effects and more comparison to other treatments. Considering this, it does appear questionable whether Ellis and Joffe Ellis (2011) can justify the claim this study validates REBT, especially considering CBT is ultimately an approach that does differ. Although Ellis and Joffe Ellis (2011) claim limited studies specifically focused on REBT exist, they can be found and do provide support for this approach. One example is the case study in which Wood (2017) provided REBT to an elite archer experiencing performance related anxiety before and during competition. During seven sessions the client received education about ''ABC theory.'' Her irrational beliefs were then ''disputed'' and after having learned about her irrational beliefs she was to test out her newly developed philosophy as ''homework''. This entailed competing in competitions in which expectation to perform well was high. [[wikipedia:Self-report_inventory|Self-report]] data showed that as her irrational belief decreased her rational beliefs improved in regularity. The results of the competitions showed improvement in scores correlating with the change in beliefs. In follow up six months later, the improved rational belief and competition scores had continued. Although it may be suggested that the self-report is questionable as the positive effects may be related to the [[wikipedia:Hawthorne_effect|Hawthorne effect]], it is likely this athlete enjoyed positive results due to REBT. Another study focused on the theory and mechanisms of change in relation to the effects of both REBT and CBT on major depressive disorder (MDD). Significant positive benefit was found for both approaches. Interestingly, in looking at the mechanisms of change, the researchers noted that restructuring irrational beliefs appeared to have a positive effect on depressed mood and automatic negative thought (Szentagotai, David, Lupu, & Cosman, 2008). As automatic negative thought is central to CBT this may confirm Albert Ellis' claim that successful CBT research also confirms the efficacy of REBT. Additionally, this study was performed in Romania, thus confirming the efficacy of REBT in a cross-cultural setting and stronger results may have been found if contributing factors, like homework compliance, was considered. It should be noted that the practitioners in this study had 7 to 14 years of clinical experience, therefore, the results may not have been as promising if clinicians with less experience were used. ==Conclusion== REBT emerged in the mid-1950s from Albert Ellis' desire to assist people to alleviate emotional difficulties such as anxiety, panic, depression, anger, and fear. Many of the foundational REBT systems were conceived by combining what Ellis believed to be the most effective historical philosophies with favoured aspects from contemporary psychology at that time. In order to change difficult emotions, the central focus for REBT is in changing thoughts which are believed to be a strong contributor to emotional difficulties that people experience. ABC theory explains that most people believe that an activating event causes the emotional response. However, perception of the event, and therefore the emotional response, is made to be more difficult by the person's thoughts about the event. Additionally, REBT asserts that thoughts, emotions, and behaviour are all interconnected. Therefore a change in one will affect the others. Considering this, REBT asserts that if one profoundly changes irrational thoughts it will have a profound effect on behaviour and emotions. Due to the combination of different perspectives used to create REBT, therapy includes a wide variety of techniques and considerations. In therapy, clients are taught ABC theory with the aim of bringing their attention to the impact of rational and irrational thinking. Therapy includes cognitive, emotive, and behavioural techniques, humour and bringing awareness to use of detrimental semantics. In order to prevent inevitable negative life experiences resulting in emotional difficulties an REBT therapist uses these techniques to teach clients to accept themselves, others, and life. As part of this, the therapist disputes clients' irrational beliefs and teaches more rational ways of thinking. Therapy also includes homework assignments that clients are expected to complete between sessions. Through the therapy process it is believed they will grow a new understanding of how to perceive life and all that it involves. After the experience of REBT, and by continuing with the hard work and practice REBT requires, it is considered that people will negotiate life's inevitable problems without the unhealthy emotional difficulties experienced previously. ==See also== * [[wikipedia:Rational_emotive_behavior_therapy|REBT]] (Wikipedia) ==References== {{Hanging indent|1= Bond, F. W., & Dryden, W. (2000). How rational beliefs and irrational beliefs affect people's inferences: An experimental investigation. ''Behavioural and Cognitive Psychotherapy, 28''(1), 33-43. Butler, A. C., Chapman, J. E., Forman, E. M., & Beck, A. T. (2006). The empirical status of cognitive-behavioral therapy: A review of meta-analyses. ''Clinical Psychology Review, 26''(1), 17-31. http://dx.doi:10.1016/j.cpr.2005.07.003 Carpintero, H. (2004). Watson's Behaviorism: A Comparison of the Two Editions (1925 and 1930). ''History of Psychology, 7''(2), 183-202. http://dx.doi:10.1037/1093-4510.7.2.183 Christopher, M. S. (2003). Albert Ellis and the Buddha: rational Soul Mates? A comparison of Rational Emotive Behaviour Therapy (REBT) and Zen Buddhism. ''Mental Health, Religion & Culture, 6''(3), 283-293. http://dx.doi:10.1080/1367467031000100975 David, D., & Szentagotai, A. (2013). Per Aspera Ad Astra: 100 years since the birth of Albert Ellis. From shadow to the mainstream. ''Journal of Cognitive and Behavioral Psychotherapies, 13''(2A), 441-443. Dryden, W. (2010). What is Rational Emotive Behaviour Therapy (REBT)?: Outlining the approach by considering the four elements of its name. ''The Rational Emotive Behaviour Therapist, 13''(1), 22-32. Dryden, W., & Neenan, M. (2004). ''Rational emotive behavioural counselling in action:'' London: Sage. Ellis, A. (1974). ''Technique of disputing irrational beliefs (DIBS).'' New York: Institute for Rational-Emotive Therapy. Ellis, A. (1999). Early theories and practices of rational emotive behavior therapy and how they have been augmented and revised during the last three decades. ''Journal of Rational-Emotive & Cognitive-Behavior Therapy, 17''(2), 69-93. Ellis, A., & Joffe Ellis, D. (2011). ''Rational Emotive Behavior Therapy''. Washington DC: American Psychological Association. Kerr, N. J. (1984). The tyranny of the should's. ''Perspectives in Psychiatric Care, 22''(2), 16-19. McWilliams, S. A. (2016). Cultivating constructivism: Inspiring intuition and promoting process and pragmatism. ''Journal of Constructivist Psychology, 29''(1), 1-29. http://dx.doi:10.1080/10720537.2014.980871 Rait, S., Monsen, J. J., & Squires, G. (2010). Cognitive behaviour therapies and their implications for applied educational psychology practice. ''Educational Psychology in Practice, 26''(2), 105-122. http://dx.doi:10.1080/02667361003768443 Szentagotai, A., David, D., Lupu, V., & Cosman, D. (2008). Rational emotive behavior therapy versus cognitive therapy versus pharmacotherapy in the treatment of major depressive disorder: Mechanisms of change analysis. ''Psychotherapy: Theory, Research, Practice, Training, 45''(4), 523-538. http://dx.doi:10.1037/a0014332 }} ==External links== * [https://www.youtube.com/watch?v=C7GyIGBxW4k Interview with Dr Debbie Jofffe Ellis, Albert's wife] (youtube) * [http://albertellis.org/rebt-in-the-context-of-modern-psychological-research/ Article about Ellis and REBT on the Albert Ellis Institute website] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Psychotherapy]] 6h97besrawa06agu45zeejznibjnc5t 10 226882 2816074 2725049 2026-06-17T05:47:20Z Jtneill 10242 Update for 2026 2816074 wikitext text/x-wiki #REDIRECT [[Template:METF/2026]] 38w9k4prtcapkugzjoroeucwxwyem1s 1 228513 2815993 2624334 2026-06-16T18:21:47Z Psubhashish 281549 /* Proposal for a proceeding */ new section 2815993 wikitext text/x-wiki {{WikiJournal discussions|Memoires-des-commissaires-du-roi-et-de-ceux-de-sa-majesté-brittanique MG 9383 (crop).tif}} {{Archive box| [[/2017]] Discussions may also take place at the <br>'''[https://groups.google.com/forum/#!forum/wikijhum/join public mailing list]''' }}[[Category:WikiJournal of Humanities]] == Adding "psychology" to main topics == As per [[Talk:WikiJournal_User_Group#WikiJournal_of_Psychology]], I think we can add "psychology" after "the humanities, arts, and social sciences" at this page. Anyone disagrees? [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 19:59, 17 January 2018 (UTC) :I agree. I had originally envisage it as a part of the social sciences remit of WikiJHum, but it might be good to be explicit. It would also be good to approach some active [[wikipedia:Wikipedia:WikiProject_Psychology|psychology Wikipedians]] and [http://psychtastic.com/2015/02/list-of-psychology-open-access-journals/ open access journal editors]. For example, [[w:User:Markworthen]], [[w:User:Famousdog]], and [[w:User:Cathrotterdam]]. I'll hold off contacting until others have given opinions. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 20:43, 17 January 2018 (UTC) :I agree as well. By being inclusive with more subject-areas will only increase the articles we receive. If that particular area really is successful, we can all discuss about the subject taking on its own Wiki Journal later. [[User:Jackiekoerner|Jackiekoerner]] ([[User talk:Jackiekoerner|discuss]] • [[Special:Contributions/Jackiekoerner|contribs]]) 13:14, 19 January 2018 (UTC) ::I just had the point made to me that members of the psychology community can often view their field as a science rather than a social science. I think in the end, the authors can submit to whichever of the journals that feel best fits their topic, and the editors can recommend the submission be moved to one of the other journals in the unlikely case that there's a clash of opinions. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:20, 23 January 2018 (UTC) :::I concur with the freedom to choose journal, e.g., social psychology at one extreme and psychometrics at the other! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 06:17, 24 January 2018 (UTC) ::::I added psychology to the [[WikiJournal_of_Humanities|official subjects of the journal]] now. Still, I agree authors may choose to submit to whichever journal they feel is most fitting. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:31, 25 January 2018 (UTC) == Publishing workflow == Hi, I'm trying to imagine how an author goes about submitting their article to the journal if it's a rewrite of a Wikipedia page. Would they first edit the Wikipedia page and then submit the page? thanks, [[User:Rachel Helps (BYU)|Rachel Helps (BYU)]] ([[User talk:Rachel Helps (BYU)|discuss]] • [[Special:Contributions/Rachel Helps (BYU)|contribs]]) 16:09, 22 January 2018 (UTC) :{{re|Rachel Helps (BYU)}} They have the option of either updating on Wikipedia then copying across to the submission page, or submitting fully new version on the the WikiJournal. We've seen people do both versions (e.g. [[Lysine: Biosynthesis, Catabolism and Roles|this lysine article]] is a complete rewrite that was directly submitted to WikiJSci, whereas [[WikiJournal of Medicine/The Cerebellum|Cerebellum]] was updated on Wikipedia before submission). The ideal is probably rewrite on Wikipedia so that other wikipedia editors get a heads up that it's being updated (especially if the Wikipedia page is [[wikipedia:Wikipedia:WikiProject_assessment#Grades|class C]] or above), but I think that either is acceptable. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 17:05, 23 January 2018 (UTC) ::Thanks, I have a clearer picture of how submissions work now. [[User:Rachel Helps (BYU)|Rachel Helps (BYU)]] ([[User talk:Rachel Helps (BYU)|discuss]] • [[Special:Contributions/Rachel Helps (BYU)|contribs]]) 16:40, 24 January 2018 (UTC) == Email address == Are we really using Submissions@WikiJMed.org for WikiJHum, as specified on this page? I can understand if a new email account/domain was felt to be unnecessary but also thought this might be a mistake. --[[Special:Contributions/2A00:23C4:C184:6700:81EA:6176:9EE0:A9BA|2A00:23C4:C184:6700:81EA:6176:9EE0:A9BA]] ([[User talk:2A00:23C4:C184:6700:81EA:6176:9EE0:A9BA|discuss]]) 11:45, 30 March 2018 (UTC) :You raise a good point. We don't have an email set up yet for WikiJHum. I'll try to get that organised asap. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:33, 31 March 2018 (UTC) ::This is indeed an important issue. I'll look into it tomorrow (it's starting to get late in Sweden now!) [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:01, 31 March 2018 (UTC) :::A contact email is now working, see [[WikiJournal of Humanities/Contact]]. I've emailed the board to discuss which people should have access and be responsible for checking emails to this address. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 12:15, 2 April 2018 (UTC) == WikiJournal of Humanities has an ISSN! == At 3:56am Sydney time today, the Library of Congress issued an ISSN for the WikiJournal of Humanities, meaning we can now publish our first issue. --[[User:Fransplace|Fransplace]] ([[User talk:Fransplace|discuss]] • [[Special:Contributions/Fransplace|contribs]]) 22:58, 22 October 2018 (UTC) == Feedback from Analysis & Policy Observatory == At the [[w:Analysis & Policy Observatory|Analysis & Policy Observatory]] Forum on "Redesigning the Public Knowledge System" there was great interest in WikiJournals, particularly WikiJHum, that I thought I'd summarise here. The [[w:Campbell_Collaboration|Campbell Collaboration]] does amazing metaanalyses of policy interventions to inform evidence-based politics and are a sister organisation the the medical [[w:Cochrane collaboration|Cochrane collaboration]]. Their CEO (Howard White) gave a great talk but sadly had to leave straight after. Several people mentioned that Wiki?JHum could be an ideal way to engage them on ensuring articles are up to date (e.g. [[w:Domestic_violence#Nonsubordination_theory|Mandatory arrest for domestic violence]]). There was interest in updating a set of properties on WikiData then publishing a WikiJournal article to describe the changes (e.g. on concept hierarchies in political theory). At the APO forum a few additional missing topics were pointed out that might make good WikiJHum articles: * [[w: Knowledge integrity|Knowledge integrity]] * [[w:Knowledge infrastructure|Knowledge infrastructure]] * [[w:public interest journalism|Public interest journalism]] The session was recorded and I believe that the videos will eventually be uploaded [https://www.youtube.com/channel/UChfOJfSWv9WaJlQKsrYI3Mg/videos here]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:23, 25 October 2018 (UTC) == Advertising template for posting to talk pages == I've put together a short template that can be used to advertise WikiJHum on WikiProject, or user talk pages: [[w:Template:WJH_advert_2018_Dec]]. Feel free to directly edit to improve it! [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 11:45, 23 November 2018 (UTC) == question about editorial board process == I just noticed that the header at [[Talk:WikiJournal of Humanities/Editorial board]] states that "Editors with at least 30 edits to WikiJournal of Humanities pages." is a requirement to participate in elections. Is that current and enforced? --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 03:23, 10 March 2019 (UTC) :{{re|Mu301}} Currently only loosely. Most votes are cast by editorial board members, so the remainder can still be checked individually. I'd be interested on your thoughts on how to robustly define an electorate that avoids gaming. Checking voters manually is currently still possible, but will become less viable as the project expands. Also, see [https://meta.wikimedia.org/wiki/WikiJournal_User_Group/Meetings/2019-04-29? this recent discussion] on Affcom feedback on defining the electorate in the bylaws. I'll also post something to the [[Talk:WikiJournal User Group]] page about it once I've drafted some proposed wording. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 00:54, 2 May 2019 (UTC) == SHERPA/RoMEO == I've submitted to the details for WikiJHum to SHERPA/RoMEO via the [http://sherpa.ac.uk/forms/new-journal.php journal submission form]. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 12:50, 2 May 2019 (UTC) == Learning and/or education related writing?== Hello folks, first up let me say how fantastic it is to see the journals developing and holding on Wikiversity. It is a significant and long needed development. I've been very absent from WV over the years the journals have been developing, mostly because my day job has taken me away from it :( I'm looking for a place to submit my writings on learning and education. It might not be up to journal standard most of the time, but I'd like to get into the workflow toward that goal. I'm just not sure if there is a place in the journals for it yet.. I hope it might be Humanities...? A sample of finished works that could easily be shaped into the journal's format: # [[User:Leighblackall/Humanist technology|Humanist technology]] - A presentation to eLearning Korea, arguing for the humanities taking a much stronger role in the consideration of technology. This project relates to the [[User:Leighblackall/An ethical framework for ubiquitous learning|An ethical framework for ubiquitous learning]]. # [[User:Leighblackall/Badges: identify talent and brand by association|Badges: identify talent and brand by association]] - A research and development project at RMIT, looking at how badges could be used to improve the employment prospects of graduates in the Advertising degree. # [[User:Leighblackall/An ethical framework for ubiquitous learning|An ethical framework for ubiquitous learning]] - Presented as work in progress to the IEEE Conference 2014 # [[User:Leighblackall/Data and Power|Data and Power]] - A presentation to the Melbourne University Analytics Forum 2014 # [[User:Leighblackall/Open Online Courses and Massively untold stories|Open Online Courses and Massively untold stories]] - A critical history of Massive Open Online Courses, published by Ascilite2014 # [[Journalism studies and Wikinews]] - A collaborative paper published in the Australian and New Zealand Communication Association conference: Communicating Change and Changing Communication in the 21st Century. 2013 # [[User:Leighblackall/Open Education Practices: A User Guide for Organisations|Open Education Practices: A User Guide for Organisations]] - A manual for the New Zealand Ministry of Education 2009 # [[Sustainability considerations relating to the use of Second Life for education]] - A critical review for the New Zealand Ministry of Education 2009 # [[User:Leighblackall/Socially constructed media and communications|Socially constructed media and communications]] - A presentation to Ascilite 2007 Other works in progress are listed on my Userpage. Looking forward to hearing from you folks --[[User:Leighblackall|Leighblackall]] ([[User talk:Leighblackall|discuss]] • [[Special:Contributions/Leighblackall|contribs]]) 05:50, 13 August 2019 (UTC) == jibberish == I can't find the right place to edit the text. Can someone familiar with the transuded templates fix "be '''ofssrnsddsdkljnfa;sdfjgnlksadjfghbn''' an appropriate open license"? [[User:Argento Surfer|Argento Surfer]] ([[User talk:Argento Surfer|discuss]] • [[Special:Contributions/Argento Surfer|contribs]]) 14:53, 21 August 2019 (UTC) : Done, thanks! —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 15:31, 21 August 2019 (UTC) :: Thank you both for catching that. The edit history indicates that it was me who introduced the error. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:57, 21 August 2019 (UTC) == WJH and Wikipedia == I didn't understand than WJH could be a place to publish encyclopedic article coming from Wikipedia. In my point of view scientific work and encyclopedic one are not really similar in term of target, methodology and epistemology. Also I was looking to WJH as place to complete a gap on scientist publication in term of transparency and free accessibility, not for working in parallel of Wikipedia for enhancing and promoting best articles. In my opinion that should be done directly on Wikipedia for keeping WJH concentrated on the publication of original research and new knowledge in general that cant be produce on wikipedia. Sorry if I came after probably a long creation process where this kind of topics was already discussed. [[User:Lionel Scheepmans|Lionel Scheepmans]] ^{<big>✉ </big> [[User talk:Lionel Scheepmans|Contact]]} _{(French native speaker)} 17:53, 31 October 2019 (UTC) :I have to agree. I'm not sure what value this adds to the wp work. The only exception that I can think of would be for WJM to review a wp page about alternative medicine with the intent to distribute that version in paper format to the public. I'm a bit disappointed that all four published articles (so far) are wp adaptations but not a single original contribution. I'd also like to point out a glaring omission: the abstracts for 2 of 4 published articles listed at [[WikiJournal of Humanities]] omit "et al" implying they are the work of a sole author. (I realilze that the pdf versions includes it.) I'm a strong supporter of the idea of a WikiJournal but I have to ask: why is this so heavely focused on wp? --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 21:59, 31 October 2019 (UTC) ::The current [[WikiJournal_of_Humanities/Publishing#Publication_formats|WikiJHum publication formats list]] includes both encyclopedic-style reviews as well as original research articles. It seems as though initially most of the articles submitted for review have been from wikipedia (via [[w:wp:JAN]]). This is likely a function of who has been informed of the journal's existence (i.e. it's best know of within the wikimedia community). [[WikiJMed]] and [[WikiJSci]] have tended to get relatively more submissions of research articles, but these have also been more widely advertised at academic conferences and with editors emailing potential authors. If WikiJHum wants to change the focus more towards research articles, one way to boost that side would be to contact potential knowledgeable authors and invite them to submit articles that can be used as exemplar works when showcasing the format to others. It could also be possible to raise changing the article formats listed (which were merely adapted form the existing formats at WikiJMed and WikiJSci) to better fit with humanities, arts and soc sci subjects. ps, It's actually the case that all four of the currently published articles in WikiJHum are adapted from WP so I've fixed the missing ''et al''s on the journal main page to match what's on the articles. I think it was an issue from an old template switchover. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 23:04, 31 October 2019 (UTC) :::Thanks for your comment [[User:Mu301|mikeu]] and for this reply [[User:Evolution and evolvability|T.Shafee]], so the actual question is how to attract researcher to Wikijournal ? I have two plans in mind. The first would be to present a first example of research at the final stage of its publication, the second would perhaps be to produce posters to be placed in universities. For the first one, I could start with a chapter of my thesis that I could translate into English. I guess French-language publications are not accepted, is that right? For the second one, I'm willing to put up posters in my university and discuss them with my dean. [[User:Lionel Scheepmans|Lionel Scheepmans]] ^{<big>✉ </big> [[User talk:Lionel Scheepmans|Contact]]} _{(French native speaker)} 23:49, 31 October 2019 (UTC) P.S. But before speaking with my dean, It should be me more comfortable for me to master both processes: first to be peer review coordinator, second to be author of a peer reviewed research or chronologically unversed. ::::(recovering from edit conflict) Please understand that this is a critique and not a criticism. The intent is to engage in a robust debate which improves the value of this endeavor. I have no problem with following through on wp article reviews that are already in the pipeline. I also recognize that this is good "practice" to refine the procedures and processes for vetting submissions. I would suggest that we reconsider inclusion of wp reviewing within the scope of all journals here. A strong case could be made that this activity is contrary to the WMF accepted project creation proposal which states that wikiversity is not: "[[Wikiversity:Wikiversity_project_proposal#What_Wikiversity_is_not|A duplication of other Wikimedia projects]]." (But I won't press that point.) In my opinion this also weakens the case for a dedicated WikiJournal project site. I can't in good conscience support a site creation proposal at this time though I've refrained from expressing a negative opinion at the meta proposal page. I consider that propsal to be premature. There is obviously a need and/or desire for a wiki based journal. I see an over-emphasis on technical minutiae but that lacks a clearly stated justification about who and how this journal serves. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 00:08, 1 November 2019 (UTC) :::::{{re|Lionel Scheepmans}} You might have to email the board re:language, however so far there's only been experience at organising peer review in English (though both language versions could be published afterwards). See the [[WikiJournal_User_Group/Potential_upcoming_articles|Potential_upcoming_articles]] of the other journals for an idea on some of the other publication formats being trialled. A the moment, WikiJHum has been focussed on review articles than can be piped into Wikipedia (I suspect as a way to kick-start initial submissions). Re: posters, there are some posters existing for WikiJSci [https://commons.wikimedia.org/wiki/File:WikiJournal_of_Science_Poster.pdf] and WikJMed [https://commons.wikimedia.org/wiki/File:WikiJournal_of_Science_Poster.pdf] that (also powerpoint link [https://drive.google.com/open?id=0B4LQzkvkbO9Yd0tTQ0NQdVV5YTA here]) that could either be adapted for WikiJHum, or something could be made from scratch (seeing as it's a different audience so likely has different priorities/aesthetic).[[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:55, 6 November 2019 (UTC) :::::{{re|Mu301}} We could add discussion of the future of the different formats to the agenda of the [[metawiki:WikiJournal_User_Group/Meetings|next meeting]]. My understanding is that you're more in favour of focussing on original research articles? [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 09:55, 6 November 2019 (UTC) ::::::Yes, I share with {{re|Mu301}} this point of view. And I think it's a good idea [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]], to add discussion of the future of the different formats to the agenda of the [[metawiki:WikiJournal_User_Group/Meetings|next meeting]]. But I've missed the last meeting and I still don't know why... Perhaps a mistake about the gathering time or with the Zoom program. I don't know. [[User:Lionel Scheepmans|Lionel Scheepmans]] ^{<big>✉ </big> [[User talk:Lionel Scheepmans|Contact]]} _{(French native speaker)} 10:58, 7 November 2019 (UTC) :::::::There should definitely be more of a focus on original content. If WJs desire to continue to publish vetted WP articles I think there should be a lengthy discussion about what the justification for this is. Who is the audience that benefits and how? --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 14:22, 7 November 2019 (UTC) ::::::::{{re|Lionel Scheepmans|Mu301}} Good idea to discuss the [[WikiJournal User Group/Publishing#Publication formats|current formats]] in more detail at the [[metawiki:WikiJournal_User_Group/Meetings|next meeting]]. I definitely agree that attracting submissions of high-quality original research is useful (indeed, it was part of the [[Talk:WikiJournal User Group/Open tasks and discussions#Free Journal Network|feedback from the Free Journal Network]]) and has started to occur at [[WikiJMed]] through concerted efforts. For encyclopedia review articles, there are a couple of additional considerations. A) Some authors will write such articles for a peer reviewed WikiJournal but would not have written directly for Wikipedia (e.g. [https://doi.org/10.15347/wjm/2018.001], [https://doi.org/10.15347/wjs/2019.004]), benefiting readers of the encyclopedia with new content that has been rigorously vetted and benefiting the authors with a citable version that can help justify the time spent (equivalent to [https://collections.plos.org/topic-pages PLOS's Topic Pages format]). B) If that [[w:wp:J2W|Journal→Wiki]] format is kept by WikiJHum, then there is a benefit to also including a [[w:wp:W2J|Wiki→Journal format]] in order to prevent potential contributors being discouraged from editing Wikipedia in order to 'save' their work for submission to a journal (i.e. wp treated as a preprint server), or those who wanted to submit, but initially drafted the content on Wikipedia (e.g. [https://doi.org/10.15347/wjs/2019.006]), (equivalent to [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4242787/ Open Medicine's Wikipedia clinical review]). There was also [[wikipedia:Wikipedia:Village_pump_(policy)/Archive_153#WP:JAN_/_WikiJournals|related discussion on this]] over at en.wp village pump earlier this year. My opinion is that there is a place for encyclopedic reviews, other review formats, and original research (some editors focus on particular favoured formats). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 07:48, 13 November 2019 (UTC) {{od}} Ok, let's talk about all of this during the next meeting. I've missed the last meeting on 30/10 when I was connected at 23:00 GMT. Did I have the wrong time or is it the computer program that didn't work? Is there a page that I can put in my follow up list to be kept informed of the exact GMT time of upcoming meetings? [[User:Lionel Scheepmans|Lionel Scheepmans]] ^{<big>✉ </big> [[User talk:Lionel Scheepmans|Contact]]} _{(French native speaker)} 13:16, 13 November 2019 (UTC) :No problem. We've tended to use a link to [https://www.timeanddate.com/worldclock/fixedtime.html?msg=WikiJournals+Combined+Meeting+October+2019&iso=20191030T19&p1=179&ah=1 timeanddate.com], but we could try a calendar link share. I think the time you listed was correct, so it must have been the zoom link. One of the agenda items is to try an open source alternative, so I think Jitsi is the most likely. I've been testing it out and it's mostly very stable (oddly the 'blur background' option causes my computer to crash). [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 22:09, 13 November 2019 (UTC) ::Hi, I started the same topic on [[Talk:WikiJournal_User_Group#Stop_publishing_Wikipedia_articles_on_WJ|Talk:WikiJournal_User_Group]] to increase its visibility. Best, [[User:Lionel Scheepmans|Lionel Scheepmans]] ^{<big>✉ </big> [[User talk:Lionel Scheepmans|Contact]]} _{(French native speaker)} 11:31, 15 February 2020 (UTC) == Application to DOAJ for WikiJHum == Given the discussion at the [[metawiki:WikiJournal_User_Group/Meetings/2020-06-12|last open meeting]], it's probably time for WikiJHum to apply to DOAJ. *[[WikiJournal User Group/Applications|Previous and current WikiJournal applications]] *[https://doaj.org/application/new DOAJ application form] *[https://doaj.org/toc/2470-6345 WikiJSci DOAJ entry example] Any volunteers to shepherd the process? [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 07:02, 14 June 2020 (UTC) == File not found: ....pdf == Hello, Why download as pdf not work on: * Æthelflæd, Lady of the Mercians * Rosetta Stone * A grammatical overview of Yolmo (Tibeto-Burman) Facing error like <code><nowiki>File not found: /v1/AUTH_mw/wikiversity-en-local-public/a/a0/AEthelfl%C3%A6d%2C_Lady_of_the_Mercians.pdf</nowiki></code> '''--'''[[User:علاء |<span style="color:black;font-family:Script MT Bold;font-size:16px;">Alaa</span> ]] [[User_talk:علاء |:)..!]] 20:27, 19 June 2020 (UTC) :{{ping|Evolution and evolvability}} can you help? '''--'''[[User:علاء |<span style="color:black;font-family:Script MT Bold;font-size:16px;">Alaa</span> ]] [[User_talk:علاء |:)..!]] 19:03, 20 June 2020 (UTC) ::{{re|علاء }} I've had a look into it and it seems to be to do with the new automated pdf naming system (files have to be name "file:[title_of_article].pdf". Now that I've updated the pdf parameters [[WikiJournal of Humanities/Volume 2 Issue 1|on the volume page]] it seems to be working again. [[User:Evolution and evolvability|T.Shafee(Evo&#65120;Evo)]]^{[[User talk:Evolution and evolvability|talk]]} 06:16, 21 June 2020 (UTC) :::{{re|Evolution and evolvability}} Thanks '''--'''[[User:علاء |<span style="color:black;font-family:Script MT Bold;font-size:16px;">Alaa</span> ]] [[User_talk:علاء |:)..!]] 09:34, 21 June 2020 (UTC) == Removing "psychology" from main topics? == With the advent of [[WikiJournal of Psychology, Psychiatry and Behavioral Sciences]], shouldn't "psychology" be removed from the description of this journal? Thanks, [[User:DavidMCEddy|DavidMCEddy]] ([[User talk:DavidMCEddy|discuss]] • [[Special:Contributions/DavidMCEddy|contribs]]) 18:11, 12 October 2021 (UTC) :@[[User:DavidMCEddy|DavidMCEddy]] Maybe wait until first volume is published, but add a note that psychology submissions are now welcome at that dedicated venue? The issue is that the new venue is not as well indexed (SCOPUS), so many authors may still prefer the main medical journal instead. [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:38, 8 November 2022 (UTC) ::<nowiki>{{re|Piotrus}}</nowiki> and {{re|[[User:DavidMCEddy|DavidMCEddy]]<nowiki>}} We anticipate the standalone psychology journal to launch in the second half of 2023. In the meantime, psychology articles can be submitted to Medicine, Medicine or Humanities depending on the topic (probably the first two, until we stabilize the submission situation in Humanities). </nowiki> [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:34, 11 November 2022 (UTC) == Checking in on submission status == Hello! I just wanted to check that [[WikiJournal Preprints/Nice state history, if you can get it: Exploring open access and digital object identifier (DOI) registration in current U.S. state history journals|the preprint]] I submitted two months ago was received as it hasn't shown up on the [[WikiJournal of Humanities/Potential upcoming articles|submission tracking page]]. I'm happy to resubmit if needed, just let me know. Thanks for your help! Kindly —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 00:07, 13 October 2021 (UTC) : Hi all, sorry to be a bother but it's been three and a half months and I just wanted to be sure the manuscript has been received. Tagging a few folks on the editorial board in case folks don't watch this page: {{u|Fransplace}}, {{u|Eystein_Thanisch}}, {{u|Smvital}}. Thank you! —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 21:37, 29 November 2021 (UTC) == Recruiting technical editors == We are hiring new [[WikiJournal User Group/Technical editors|technical editors]] for the journals. Please see [https://www.linkedin.com/posts/andrewcleung_technical-editor-job-poster-activity-6912636772371828736-LteF this job posting for details.] [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:24, 29 March 2022 (UTC) == Proposal to introduce "Inactivity removal policy" to the bylaws == There is an ongoing discussion to propose introducing an inactivity removal policy for editorial board members. Full details [[Talk:WikiJournal User Group#Proposal to introduce "Inactivity removal policy" to the bylaws|can be viewed here]]. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:23, 12 September 2022 (UTC) == [[WikiJournal of Humanities/Potential upcoming articles]] - update needed or no reviewers for years? == I glanced at that page. It seems we have submissions that didn't get any reviewers since 2019??? [[User:Piotrus|Piotrus]] ([[User talk:Piotrus|discuss]] • [[Special:Contributions/Piotrus|contribs]]) 03:36, 8 November 2022 (UTC) == Volume 5 or 6 == Great to see that "[[WikiJournal of Humanities/Loveday, 1458|Loveday, 1458]]" has now been published! Its [https://search.crossref.org/?q=10.15347%2FWJH%2F2023.001&from_ui=yes metadata] lists it in volume 6 of the ''WJH'' but the [[WikiJournal of Humanities|journal homepage]] lists it as volume 5. I suspect this discrepancy arises from no articles being published in 2022 -- do we skip volume 5 or correct "Loveday"'s info to indicate it's part of volume 5? —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 21:33, 23 June 2023 (UTC) :It should be volume 5. We will make corrections to the metadata. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 04:35, 29 June 2023 (UTC) == Original research structure == Is the traditional ''Introduction'', ''Methods'', ''Analysis'', ''Results'', and ''Discussion'' structure possible? I don't see any examples of it under WJH. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:52, 5 September 2023 (UTC) :@[[User:Juandev|Juandev]] You're probably correct. This structure makes very little sense for humanities articles, although social science original research often use the structure that are more commonly found in science and medical fields. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 03:09, 19 September 2023 (UTC) ::Well it is important to determine it before writing. You usually write articles on a expected design of the Journal. Not just social science, but also information or data science may favorize clear paragraph naming. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:20, 22 September 2023 (UTC) == Article revisions done == Im not entirely sure where to post this but I did finish the revisions for [[WikiJournal Preprints/The Holocaust in Slovakia]]. As I discussed with Ohana, I didn't expand the scope of the article to include Romani people, and I was unable to implement some of reviewer #2's comments because the information that would clarify is not in the cited source, or any other source that I'm aware of. [[User:Buidhe|Buidhe]] ([[User talk:Buidhe|discuss]] • [[Special:Contributions/Buidhe|contribs]]) 17:57, 25 February 2024 (UTC) == Concerning current self-imposed threats to humanity? == Apologies if I am intruding, but what about this subject? I do not see any of the editions relating to this. [[User:Janosabel|Janosabel]] ([[User talk:Janosabel|discuss]] • [[Special:Contributions/Janosabel|contribs]]) 10:14, 1 May 2024 (UTC) == Proposal for a proceeding == Dear colleagues at WJH, I'm writing to you to propose a proceeding for our upcoming workshop series titled, "OpenSpeaks Community language documentation and archiving training". It will run from 18 May till 3 September with seven online sessions and an in-person event. The workshop will be based on a [[m:OpenSpeaks/Community_language_documentation_and_archiving_training|curriculum]], an OER. Each online session will focus on one of the [[m:OpenSpeaks/Community_language_documentation_and_archiving_training#What_participants_will_learn|seven]] identified topics. The presentation will be recorded and published as an OER, while the in-person workshop will be more hands-on, creating/improving OERs and [[m:OpenSpeaks/Tools|tools]]. Here's a brief list of tentative articles: 1. [[m:OpenSpeaks/Community_language_documentation_and_archiving_training#What_participants_will_learn|Event curriculum]] (seven chapters of the curriculum can become an article of their own, these will be co-authored by the trainers) 2. A [[m:OpenSpeaks/Oral_Knowledge_Framework|framework]] on community-based oral knowledge documentation (already peer-reviewed at Wiki Workshop 2026) 3. [[m:OpenSpeaks/Captioning|Text style guide]] inspired from EU's EBU-TT-D and BBC's style guide 4. Current [[WikiJournal Preprints/OpenSpeaks: Open Toolkit for Multimedia Documentation of Indigenous Languages|article]] in pre-print which is also a learning resource 5. Each language we've documented has multiple audio and video files. We can explore creating media-focused articles for each one. [[User:Psubhashish|Psubhashish]] ([[User talk:Psubhashish|discuss]] • [[Special:Contributions/Psubhashish|contribs]]) 18:21, 16 June 2026 (UTC) 4pi5j2jsull4im5y22sbk1fx1xuy03t 8 236683 2816013 2598202 2026-06-16T19:44:13Z Johannnes89 618724 update per enwiki 2816013 javascript text/javascript // See [[mw:Reference Tooltips]] // Source https://en.wikipedia.org/wiki/MediaWiki:Gadget-ReferenceTooltips.js /*eslint space-in-parens: ["error", "always"], array-bracket-spacing: ["error", "always"]*/ ( function () { // If you're loading the script from another wiki and want to set your settings, do that in `window` // properties with `rt_` prefix, e.g. // window.rt_REF_LINK_SELECTOR = '...'; // They will be used instead of enwiki detaults. var REF_LINK_SELECTOR = window.rt_REF_LINK_SELECTOR || '.reference, a[href^="#CITEREF"]', COMMENTED_TEXT_CLASS = window.rt_COMMENTED_TEXT_CLASS || 'rt-commentedText', COMMENTED_TEXT_SELECTOR = ( window.rt_COMMENTED_TEXT_SELECTOR || ( COMMENTED_TEXT_CLASS ? '.' + COMMENTED_TEXT_CLASS + ', ' : '' ) + 'abbr[title]' ); if ( mw.messages.get( 'rt-settings' ) === null ) { mw.messages.set( { 'rt-settings': 'Reference Tooltips settings', 'rt-enable-footer': 'Enable Reference Tooltips', 'rt-settings-title': 'Reference Tooltips', 'rt-save': 'Save', 'rt-enable': 'Enable Reference Tooltips', 'rt-activationMethod': 'Show a tooltip when I\'m', 'rt-hovering': 'hovering a reference', 'rt-clicking': 'clicking a reference', 'rt-delay': 'Delay before the tooltip appears (in milliseconds)', 'rt-tooltipsForComments': 'Show the tooltip over <span title="Tooltip example" class="' + ( COMMENTED_TEXT_CLASS || 'rt-commentedText' ) + '" style="border-bottom: 1px dotted; cursor: help;">text with a dotted underline</span> in Reference Tooltips style (allows to see such tooltips on devices with no mouse support)', 'rt-disabledNote': 'You can re-enable Reference Tooltips using a link in the footer of the page.', 'rt-done': 'Done', 'rt-enabled': 'Reference Tooltips are enabled' } ); } // "Global" variables var SECONDS_IN_A_DAY = 60 * 60 * 24, CLASSES = { FADE_IN_DOWN: 'rt-fade-in-down', FADE_IN_UP: 'rt-fade-in-up', FADE_OUT_DOWN: 'rt-fade-out-down', FADE_OUT_UP: 'rt-fade-out-up' }, IS_TOUCHSCREEN = 'ontouchstart' in document.documentElement, // Quite a rough check for mobile browsers, a mix of what is advised at // https://stackoverflow.com/a/24600597 (sends to // https://developer.mozilla.org/en-US/docs/Browser_detection_using_the_user_agent) // and https://stackoverflow.com/a/14301832 IS_MOBILE = /Mobi|Android/i.test( navigator.userAgent ) || typeof window.orientation !== 'undefined', CLIENT_NAME = $.client.profile().name, settingsString, settings, enabled, delay, activatedByClick, tooltipsForComments, cursorWaitCss, windowManager, $teleportTarget, $body = $( document.body ), $window = $( window ), $overlay = $( '<div>' ) .addClass( 'rt-overlay' ) .appendTo( $body ); // Can't use before https://phabricator.wikimedia.org/T369880 is resolved // mw.loader.using( 'mediawiki.page.ready' ).then( function ( require ) { // $teleportTarget = $( require( 'mediawiki.page.ready' ).teleportTarget ); // $overlay.appendTo( $teleportTarget ); // } ); function rt( $content ) { // Popups gadget if ( window.pg ) { return; } var teSelector, settingsDialogOpening = false; function setSettingsCookie() { mw.cookie.set( 'RTsettings', ( Number( enabled ) + '|' + delay + '|' + Number( activatedByClick ) + '|' + Number( tooltipsForComments ) ), { path: '/', expires: 90 * SECONDS_IN_A_DAY, prefix: '' } ); } function enableRt() { enabled = true; setSettingsCookie(); $( '.rt-enableItem' ).remove(); rt( $content ); mw.notify( mw.msg( 'rt-enabled' ) ); } function disableRt() { $content.find( teSelector ).removeClass( 'rt-commentedText' ).off( '.rt' ); $body.off( '.rt' ); $window.off( '.rt' ); } function addEnableLink() { // #footer-places – Vector // #f-list – Timeless, Monobook, Modern // parent of #footer li – Cologne Blue var $footer = $( '#footer-places, #f-list' ); if ( !$footer.length ) { $footer = $( '#footer li' ).parent(); } if ( !$footer.find( '.rt-enableItem' ).length ) { $footer.append( $( '<li>' ) .addClass( 'rt-enableItem' ) .append( $( '<a>' ) .text( mw.msg( 'rt-enable-footer' ) ) .attr( 'href', '#' ) .click( function ( e ) { e.preventDefault(); 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} if ( activatedByClick ) { events = { 'click.rt': onStartEvent }; // Adds an ability to see tooltips for links if ( this.type === 'commentedText' && ( this.$element.closest( 'a' ).length || this.$element.has( 'a' ).length ) ) { events[ 'contextmenu.rt' ] = onStartEvent; } } else { events = { 'mouseenter.rt': onStartEvent, 'mouseleave.rt': onEndEvent }; } this.$element.on( events ); this.hideRef = function ( immediately ) { clearTimeout( te.showTimer ); if ( this.type === 'commentedText' ) { this.$element.attr( 'title', this.comment ); } if ( this.tooltip && this.tooltip.isPresent ) { if ( activatedByClick || immediately ) { this.tooltip.hide(); } else { this.hideTimer = setTimeout( function () { te.tooltip.hide(); }, 200 ); } } else if ( this.$ref && this.$ref.hasClass( 'rt-target' ) ) { this.$ref.removeClass( 'rt-target' ); if ( activatedByClick ) { $body.off( 'click.rt touchstart.rt', this.onBodyClick ); } } }; this.showRef = function ( $element, ePageX, ePageY ) { // Popups gadget if ( window.pg ) { disableRt(); return; } if ( this.tooltip && !this.tooltip.$content.length ) { return; } var tooltipInitiallyPresent = this.tooltip && this.tooltip.isPresent; function reallyShow() { var viewportTop, refOffsetTop, teHref; if ( !te.$ref && !te.comment ) { teHref = te.type === 'supRef' ? te.$element.find( 'a' ).attr( 'href' ) : te.$element.attr( 'href' ); // harvardRef te.$ref = teHref && $( '#' + $.escapeSelector( teHref.slice( 1 ) ) ); if ( !te.$ref || !te.$ref.length || !te.$ref.text() ) { te.noRef = true; return; } } if ( !tooltipInitiallyPresent && !te.comment ) { viewportTop = $window.scrollTop(); refOffsetTop = te.$ref.offset().top; if ( !activatedByClick && viewportTop < refOffsetTop && viewportTop + $window.height() > refOffsetTop + te.$ref.height() && // There can be gadgets/scripts that make references horizontally scrollable. $window.width() > te.$ref.offset().left + te.$ref.width() ) { // Highlight the reference itself te.$ref.addClass( 'rt-target' ); return; } } if ( !te.tooltip ) { te.tooltip = new Tooltip( te ); 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} else { setTimeout( function () { $body.on( 'click.rt touchstart.rt', te.onBodyClick ); }, 0 ); } } if ( activatedByClick || tooltipInitiallyPresent ) { reallyShow(); } else { this.showTimer = setTimeout( reallyShow, delay ); } }; this.onBodyClick = function ( e ) { if ( !te.tooltip && !( te.$ref && te.$ref.hasClass( 'rt-target' ) ) ) { return; } var $current = $( e.target ); function contextMatchesParameter( parameter ) { return this === parameter; } // The last condition is used to determine cases when a clicked tooltip is the current // element's tooltip or one of its descendants while ( $current.length && ( !$current.hasClass( 'rt-tooltip' ) || !$current.data( 'tooltip' ) || !$current.data( 'tooltip' ).upToTopParent( contextMatchesParameter, [ te.tooltip ], true ) ) ) { $current = $current.parent(); } if ( !$current.length ) { te.hideRef(); } }; this.onWindowResize = function () { te.tooltip.calculatePosition(); }; } function Tooltip( te ) { function openSettingsDialog() { var settingsDialog, settingsWindow; 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} ); this.enableField = new OO.ui.FieldLayout( this.enableCheckbox, { label: mw.msg( 'rt-enable' ), align: 'inline', classes: [ 'rt-enableField' ] } ); this.hoverOption = new OO.ui.RadioOptionWidget( { label: mw.msg( 'rt-hovering' ) } ); this.clickOption = new OO.ui.RadioOptionWidget( { label: mw.msg( 'rt-clicking' ) } ); this.activationMethodSelect = new OO.ui.RadioSelectWidget( { items: [ this.hoverOption, this.clickOption ] } ); this.activationMethodSelect.selectItem( activatedByClick ? this.clickOption : this.hoverOption ); this.activationMethodSelect.on( 'choose', function ( item ) { dialog.delayInput.setDisabled( item === dialog.clickOption ); } ); this.activationMethodField = new OO.ui.FieldLayout( this.activationMethodSelect, { label: mw.msg( 'rt-activationMethod' ), align: 'top' } ); this.delayInput = new OO.ui.NumberInputWidget( { input: { value: delay }, step: 50, min: 0, max: 5000, disabled: activatedByClick, classes: [ 'rt-numberInput' ] } ); this.delayField = new OO.ui.FieldLayout( this.delayInput, { label: mw.msg( 'rt-delay' ), align: 'top' } ); this.tooltipsForCommentsCheckbox = new OO.ui.CheckboxInputWidget( { selected: tooltipsForComments } ); this.tooltipsForCommentsField = new OO.ui.FieldLayout( this.tooltipsForCommentsCheckbox, { label: new OO.ui.HtmlSnippet( mw.msg( 'rt-tooltipsForComments' ) ), align: 'inline', classes: [ 'rt-tooltipsForCommentsField' ] } ); new TooltippedElement( this.tooltipsForCommentsField.$element.find( '.' + ( COMMENTED_TEXT_CLASS || 'rt-commentedText' ) ) ); this.fieldset = new OO.ui.FieldsetLayout(); this.fieldset.addItems( [ this.enableField, this.activationMethodField, this.delayField, this.tooltipsForCommentsField ] ); this.panelSettings = new OO.ui.PanelLayout( { padded: true, expanded: false } ); this.panelSettings.$element.append( this.fieldset.$element ); this.panelDisabled = new OO.ui.PanelLayout( { padded: true, expanded: false } ); this.panelDisabled.$element.append( $( '<table>' ) .addClass( 'rt-disabledHelp' ) .append( $( '<tr>' ).append( $( '<td>' ).append( $( '<img>' ).attr( 'src', 'https://upload.wikimedia.org/wikipedia/commons/c/c0/MediaWiki_footer_link_ltr.svg' ) ), $( '<td>' ) .addClass( 'rt-disabledNote' ) .text( mw.msg( 'rt-disabledNote' ) ) ) ) ); this.stackLayout = new OO.ui.StackLayout( { items: [ this.panelSettings, this.panelDisabled ] } ); this.$body.append( this.stackLayout.$element ); }; SettingsDialog.prototype.getSetupProcess = function ( data ) { return SettingsDialog.parent.prototype.getSetupProcess.call( this, data ) .next( function () { this.stackLayout.setItem( this.panelSettings ); this.actions.setMode( 'main' ); }, this ); }; SettingsDialog.prototype.getActionProcess = function ( action ) { var dialog = this; if ( action === 'save' ) { return new OO.ui.Process( function () { var newDelay = Number( dialog.delayInput.getValue() ); enabled = dialog.enableCheckbox.isSelected(); if ( newDelay >= 0 && newDelay <= 5000 ) { delay = newDelay; } activatedByClick = dialog.clickOption.isSelected(); tooltipsForComments = dialog.tooltipsForCommentsCheckbox.isSelected(); setSettingsCookie(); if ( enabled ) { dialog.close(); disableRt(); rt( $content ); } else { dialog.actions.setMode( 'disabled' ); dialog.stackLayout.setItem( dialog.panelDisabled ); disableRt(); addEnableLink(); } } ); } else if ( action === 'deactivated' ) { dialog.close(); } return SettingsDialog.parent.prototype.getActionProcess.call( this, action ); }; SettingsDialog.prototype.getBodyHeight = function () { return this.stackLayout.getCurrentItem().$element.outerHeight( true ); }; tooltip.upToTopParent( function adjustRightAndHide() { if ( this.isPresent ) { if ( this.$element[ 0 ].style.right ) { this.$element.css( 'right', '+=' + ( window.innerWidth - $window.width() ) ); } this.te.hideRef( true ); } } ); if ( !windowManager ) { windowManager = new OO.ui.WindowManager(); $body.append( windowManager.$element ); } settingsDialog = new SettingsDialog(); windowManager.addWindows( [ settingsDialog ] ); settingsWindow = windowManager.openWindow( settingsDialog ); settingsWindow.opened.then( function () { settingsDialogOpening = false; } ); settingsWindow.closed.then( function () { windowManager.clearWindows(); } ); } var tooltip = this; // This variable can change: one tooltip can be called from a harvard-style reference link // that is put into different tooltips this.te = te; switch ( this.te.type ) { case 'supRef': this.id = 'rt-' + this.te.$originalElement.attr( 'id' ); this.$content = this.te.$ref .contents() .filter( function ( i ) { var $this = $( this ); if ( $this.hasClass( 'mw-subreference-list' ) ) { return false; } return ( this.nodeType === Node.TEXT_NODE || !( // `a[href^="#cite_ref-"]` is for Wiktionary and possibly other // sites (not English Wikipedia) where the output of the Cite // extension is slightly different $this.is( '.mw-cite-backlink, a[href^="#cite_ref-"]' ) || ( i === 0 && // Template:Cnote, Template:Note ( $this.is( 'b' ) || // Template:Note_label $this.is( 'a' ) && $this.attr( 'href' ).indexOf( '#ref' ) === 0 ) ) ) ); } ) .clone( true ); const $ol = this.te.$ref.closest( 'ol' ); if ( $ol.hasClass( 'mw-subreference-list' ) ) { this.$content = $( '<div>' ).append( $ol.siblings( '.reference-text' ).clone( true ) .css( { display: 'block', 'margin-bottom': '0.7em' } ), this.$content ); } break; case 'harvardRef': this.id = 'rt-' + this.te.$originalElement.closest( 'li' ).attr( 'id' ); this.$content = this.te.$ref .clone( true ) .removeAttr( 'id' ); break; case 'commentedText': this.id = 'rt-' + String( Math.random() ).slice( 2 ); this.$content = $( document.createTextNode( this.te.comment ) ); break; } if ( !this.$content.length ) { return; } this.isInsideWindow = Boolean( this.te.$element.closest( '.oo-ui-window' ).length ); this.$element = $( '<div>' ) .addClass( 'rt-tooltip' ) .attr( 'id', this.id ) .attr( 'role', 'tooltip' ) .data( 'tooltip', this ); var $hoverArea = $( '<div>' ) .addClass( 'rt-hoverArea' ) .appendTo( this.$element ); var $scroll = $( '<div>' ) .addClass( 'rt-scroll' ) .appendTo( $hoverArea ); this.$content = this.$content .wrapAll( '<div>' ) .parent() .addClass( 'rt-content' ) .addClass( 'mw-parser-output' ) .appendTo( $scroll ); if ( !activatedByClick ) { this.$element .on( 'mouseenter linkPopupHover', function ( e ) { if ( !tooltip.disappearing || e.type === 'linkPopupHover' ) { tooltip.upToTopParent( function () { this.show(); } ); } } ) .on( 'mouseleave', function ( e ) { // https://stackoverflow.com/q/47649442 workaround. Relying on relatedTarget // alone has pitfalls: when alt-tabbing, relatedTarget is empty too if ( CLIENT_NAME !== 'chrome' || ( !e.originalEvent || e.originalEvent.relatedTarget !== null || !tooltip.clickedTime || $.now() - tooltip.clickedTime > 50 ) ) { tooltip.upToTopParent( function () { this.te.hideRef(); } ); } } ) .click( function () { tooltip.clickedTime = $.now(); } ); } if ( !this.isInsideWindow ) { $( '<a>' ) .addClass( 'rt-settingsLink' ) .attr( 'role', 'button' ) .attr( 'href', '#' ) .attr( 'title', mw.msg( 'rt-settings' ) ) .click( function ( e ) { e.preventDefault(); if ( settingsDialogOpening ) { return; } settingsDialogOpening = true; if ( mw.loader.getState( 'oojs-ui' ) !== 'ready' ) { if ( cursorWaitCss ) { cursorWaitCss.disabled = false; } else { cursorWaitCss = mw.util.addCSS( 'body { cursor: wait; }' ); } } mw.loader.using( [ 'oojs', 'oojs-ui' ], openSettingsDialog ); } ) .prependTo( this.$content ); } // Tooltip tail element is inside tooltip content element in order for the tooltip // not to disappear when the mouse is above the tail this.$tail = $( '<div>' ) .addClass( 'rt-tail' ) .prependTo( this.$element ); this.disappearing = false; this.show = function () { this.disappearing = false; clearTimeout( this.te.hideTimer ); clearTimeout( this.te.removeTimer ); this.$element .removeClass( CLASSES.FADE_OUT_DOWN ) .removeClass( CLASSES.FADE_OUT_UP ); if ( !this.isPresent ) { $overlay.append( this.$element ); } this.isPresent = true; }; this.hide = function () { var tooltip = this; tooltip.disappearing = true; if ( tooltip.$element.hasClass( 'rt-tooltip-above' ) ) { tooltip.$element .removeClass( CLASSES.FADE_IN_DOWN ) .addClass( CLASSES.FADE_OUT_UP ); } else { tooltip.$element .removeClass( CLASSES.FADE_IN_UP ) .addClass( CLASSES.FADE_OUT_DOWN ); } tooltip.te.removeTimer = setTimeout( function () { if ( tooltip.isPresent ) { tooltip.$element.detach(); tooltip.$tail.css( 'left', '' ); if ( activatedByClick ) { $body.off( 'click.rt touchstart.rt', tooltip.te.onBodyClick ); } $window.off( 'resize.rt', tooltip.te.onWindowResize ); tooltip.isPresent = false; } }, 200 ); }; this.calculatePosition = function ( ePageX, ePageY ) { var teElement, teOffsets, teOffset, targetTailOffsetX, tailLeft; this.$tail.css( 'left', '' ); teElement = this.te.$element.get( 0 ); if ( ePageX !== undefined ) { targetTailOffsetX = ePageX; teOffsets = ( teElement.getClientRects && teElement.getClientRects() ) || teElement.getBoundingClientRect(); if ( teOffsets.length > 1 ) { for ( var i = teOffsets.length - 1; i >= 0; i-- ) { if ( ePageY >= Math.round( $window.scrollTop() + teOffsets[ i ].top ) && ePageY <= Math.round( $window.scrollTop() + teOffsets[i].top + teOffsets[ i ].height ) ) { teOffset = teOffsets[ i ]; } } } } if ( !teOffset ) { teOffset = ( teElement.getClientRects && teElement.getClientRects()[ 0 ] ) || teElement.getBoundingClientRect(); } teOffset = { top: $window.scrollTop() + teOffset.top, left: $window.scrollLeft() + teOffset.left, width: teOffset.width, height: teOffset.height }; if ( !targetTailOffsetX ) { targetTailOffsetX = teOffset.left + ( teOffset.width / 2 ); } // Value of `left` in `.rt-tooltip-above .rt-tail` var defaultTailLeft = 19; // Value of `width` in `.rt-tail` var tailSideWidth = 13; // We tilt the square 45 degrees, so we need square root to calculate the distance. var tailWidth = tailSideWidth * Math.SQRT2; var tailHeight = tailWidth / 2; var tailCenterDelta = tailSideWidth + 1 - ( tailWidth / 2 ); var tooltip = this; var getTop = function ( isBelow ) { var delta = isBelow ? teOffset.height + tailHeight : -tooltip.$element.outerHeight() - tailHeight + 1; return teOffset.top + delta; }; this.$element.css( { top: getTop(), left: targetTailOffsetX - defaultTailLeft - tailCenterDelta, right: '' } ); // Is it squished against the right side of the page? if ( this.$element.offset().left + this.$element.outerWidth() > $window.width() - 1 ) { this.$element.css( { left: '', right: 0 } ); tailLeft = targetTailOffsetX - this.$element.offset().left - tailCenterDelta; } // Is a part of it above the top of the screen? if ( teOffset.top < this.$element.outerHeight() + $window.scrollTop() + tailHeight ) { this.$element .removeClass( 'rt-tooltip-above' ) .addClass( 'rt-tooltip-below' ) .addClass( CLASSES.FADE_IN_UP ) .css( { top: getTop( true ) } ); if ( tailLeft ) { this.$tail.css( 'left', ( tailLeft + tailSideWidth ) + 'px' ); } } else { this.$element .removeClass( 'rt-tooltip-below' ) .addClass( 'rt-tooltip-above' ) .addClass( CLASSES.FADE_IN_DOWN ) // A fix for cases when a tooltip shown once is then wrongly positioned when it // is shown again after a window resize. .css( { top: getTop() } ); if ( tailLeft ) { this.$tail.css( 'left', tailLeft + 'px' ); } } }; // Run some function for all the tooltips up to the top one in a tree. Its context will be // the tooltip, while its parameters may be passed to Tooltip.upToTopParent as an array // in the second parameter. If the third parameter passed to ToolTip.upToTopParent is true, // the execution stops when the function in question returns true for the first time, // and ToolTip.upToTopParent returns true as well. this.upToTopParent = function ( func, parameters, stopAtTrue ) { var returnValue, currentTooltip = this; do { returnValue = func.apply( currentTooltip, parameters ); if ( stopAtTrue && returnValue ) { break; } } while ( ( currentTooltip = currentTooltip.parent ) ); if ( stopAtTrue ) { return returnValue; } }; } if ( !enabled ) { addEnableLink(); return; } teSelector = REF_LINK_SELECTOR; if ( tooltipsForComments ) { teSelector += ', ' + COMMENTED_TEXT_SELECTOR; } $content.find( teSelector ).each( function () { new TooltippedElement( $( this ) ); } ); } settingsString = mw.cookie.get( 'RTsettings', '' ); if ( settingsString ) { settings = settingsString.split( '|' ); enabled = Boolean( Number( settings[ 0 ] ) ); delay = Number( settings[ 1 ] ); activatedByClick = Boolean( Number( settings[ 2 ] ) ); // The forth value was added later, so we provide for a default value. See comments below // for why we use "IS_TOUCHSCREEN && IS_MOBILE". tooltipsForComments = settings[ 3 ] === undefined ? IS_TOUCHSCREEN && IS_MOBILE : Boolean( Number( settings[ 3 ] ) ); } else { enabled = true; delay = 200; // Since the mobile browser check is error-prone, adding IS_MOBILE condition here would probably // leave cases where a user interacting with the browser using touches doesn't know how to call // a tooltip in order to switch to activation by click. Some touch-supporting laptop users // interacting by touch (though probably not the most popular use case) would not be happy too. activatedByClick = IS_TOUCHSCREEN; // Arguably we shouldn't convert native tooltips into gadget tooltips for devices that have // mouse support, even if they have touchscreens (there are laptops with touchscreens). // IS_TOUCHSCREEN check here is for reliability, since the mobile check is prone to false // positives. tooltipsForComments = IS_TOUCHSCREEN && IS_MOBILE; } mw.hook( 'wikipage.content' ).add( rt ); }() ); a43nofchz87jsl0z00dy2djz6ctocgi 0 254919 2816026 2550444 2026-06-16T21:59:00Z Jtneill 10242 removed [[Category:Motivation and emotion/Book/Therapy]]; added [[Category:Motivation and emotion/Book/Psychotherapy]] using [[Help:Gadget-HotCat|HotCat]] 2816026 wikitext text/x-wiki {{title|Glasser’s Choice Theory:<br>What is choice theory and how can it be applied to improve motivation and emotion?}} <!-- {{MECR3|1=https://www.youtube.com/watch?v=y-UUEs4rb38}} --> __TOC__ ==Overview== Consider the following statements: ''“My job is making me depressed.” “This bad weather has ruined my whole day.” “I won’t be happy until my partner starts behaving differently.” “I’m never going to get ahead in life because nothing goes my way.”'' How often might you find yourself saying something similar? Engaging in negative thought patterns and behaviours can leave us feeling depressed and unmotivated. Placing blame for our emotions and behaviours on external sources that we can’t control limits our capacity to change anything. [[wikipedia:Glasser's_choice_theory|Glasser's Choice Theory]] provides a framework in which individuals can understand that all thoughts and behaviours are chosen and the only person we can control is ourselves. By understanding and applying the principles of Choice Theory individuals can learn to develop an internal [[wikipedia:Locus_of_control|locus of control,]] change behaviours and emotions and improve their lives. {{RoundBoxTop|theme=2}} ;Focus questions: # What is Glasser's Choice Theory? # What are the core concepts of Choice Theory? # What is Reality Therapy? # Do research findings support the application of Choice Theory? {{RoundBoxBottom}} ==What is Glasser's Choice Theory?== [[File:WilliamGlasser.jpg|thumb|''Figure 1.'' Glasser at the 2009 Evolution of Psychotherapy Conference.]] Choice Theory is a universal framework developed by American Psychiatrist Dr. William Glasser (1925-2013) which explains motivation and human behaviour (Robey, Burdenski, Britzman, Crowell, & Cisse, 2011). In the early 1960’s{{gr}}, Glasser began to develop Choice Theory during psychiatric residency at The University of California, Los Angeles. This was in response to what he believed were inadequacies in the psychoanalytic approach of the time (Bradley, 2014). Glasser believed the long-lasting changes his patients required could not be achieved through insight alone. He emphasised that in order to make changes and lead more effective lives, patients should cease putting blame on others and take total personal responsibility for their behaviour. Detrimental behaviours become less frequent and intense and as a consequence, interpersonal relationships are strengthened and overall life satisfaction is increased (Robey et al., 2011). Choice Theory offers an alternative to external control psychology, which currently dominates human thinking and reasoning. In external control psychology, individuals believe they have no control over what happens and that external variables such as other poeple{{sp}} or uncontrollable events are responsible for their unhappiness (Wubbolding et al. 2004). Glasser blamed much of human distress on external control psychology, stating that the forceful and punishing way in which people go about trying to control others is destructive to their relationships and results in disconnection from the people they care about. Glasser believed that disconnectedness was the reason for almost all dysfunction and psychopathology (The Glasser Institute for Choice Therapy, 2019). Glasser published his first book ''Reality Therapy'' in 1965 (as cited in Wubbolding et al. 2004) and continued to practice, teach and refine Choice Theory until his death in 2013. Although the concepts of Choice Theory were established in the first publication, the phrase “Choice Theory” was not recognised until the realease{{sp}} of Glasser's 1998 book, ''Choice Theory: A New Psychology of Personal Freedom'' (as cited in Wubbolding et al., 2014). This is the primary text used for all that is currently taught by [https://wglasser.com/ The Glasser Institute for Choice Theory]. The institute, which was developed by Glasser over 50 years ago provides learning and certification of Choice Theory to both individuals and therapists (The Glasser Institute for Choice Theory, 2019a). ==What are the core concepts of Choice Theory?== {{expand}} ===Basic needs=== [[File:Maslow's hierarchy of needs.png|thumb|''Figure 2.'' The basic needs in Choice Theory differ from Maslow's Hierarchy of needs.]] Glasser proposed that our life choices are driven internally by five genetically encoded basic needs: #Survival #Love and belonging # Power # Freedom # Fun Choice Theory suggests that all human behaviour is purposely driven, either consciously or unconsciously by the choices we make. Every choice we make, whether functional or dysfunctional is motivated by our best attempt to satisfy one or more of our basic needs and get what we want (Bradley, 2014). Unlike [[Maslow's hierarchy of needs|Maslow's]] framework (Fig 2.) the basic needs in Choice Theory are not hierarchical. They can change over time and circumstance and between individuals (Bradley, 2014). Our needs are often satisfied through our relationships with other people and by the things which bring us closer to our ideal life (Robey et al., 2011). These needs are satisfied through relationships with others and through the things that lend value to people’s lives. ==== Survival ==== Survival is the only need that is experienced universally among all creatures. It represents our need for self-preservation and reproduction (can drive sexual desire). When a person finds themselves in a situation where their safety or survival is threatened or perceived to be threatened, they will experience a physiological response. This is often characterised by activation of the [[wikipedia:Fight-or-flight_response|fight-or-flight response]] and release of the stress hormone [[wikipedia:Cortisol|cortisol]] by the mid-brain. When the body is focused actions necessary for survival, energy is taken from elsewhere. For example, reproduction, digestion and bladder control are disrupted and this can lead to negative effects such as indigestion, weakened immunity, cardiovascular issues, depression and/or anxiety (Marlatt, 2014). ==== Love and belonging ==== Because we rely on other people to meet our other needs, Glasser considered this to be the primary psychological need. Biologically, when we experience feelings of love, the brain releases dopamine similarly to when we are eating food. Because our survival depends on food and the brain doesn't differentiate, love is perceived to be just as important to our survival (Marlatt, 2014). Love can be sexual, romantic or platonic (friendship). According to Choice Theory, power or the desire to control others can derail our love relationships. Glasser stated that to keep love going between two people, the friendship component must not be lost because people are much less likely to experience feelings of ownership over their friends (Bradley, 2014). Humans are social by nature and our need to feel valued and accepted within our community, family or among peers is also biologically driven (Marlatt, 2014). ==== Power ==== The need for power has both negative and positive components. Positively, it represents feelings of accomplishment, competence, achievement, respect, success and recognition. Negatively it it can manifest in two extremes. Because the primary human need is love and belonging, people make try to exert power over their partner in romantic relationships. The switch back to external control can result in selfish, controlling or manipulative behaviours which will ultimately lead to disconnection. Either the pursuit for power over others or experiencing oneself as powerless may cause unhappiness and result in the need for counselling. Because of our innate need to experience feelings of power, we will experience distress if we don't have power over ourselves (Bradley, 2014). ==== Freedom ==== This is our need to be independent, autonomous and able to express ourselves creatively. When we perceive that our freedom is threatened or we are unable to express ourselves we may choose to channel our creativity in a destructive manner (Bradley, 2014). ==== Fun ==== Glasser{{fact}} stated that our need for fun is linked to learning through play. Ultimately if we are not having fun or feeling like we are enjoying life, we are most likely not going to fell{{sp}} happy or fulfilled. We may seek out unhealthy behaviours instead to try to fulfill the need, such as substance abuse (Bradley, 2014). ===Quality world pictures=== Quality world pictures are the individually unique mental representations of everything that we ideally want. The quality world we hold in our minds represents the people, ideas and things that upon having would fulfill all of our needs. Unlike our needs, our quality world is shaped after birth through our influences and experiences. It continuously changes throughout life as we move through different stages of development and experience (Walter, Lambie & Ngazimbi, 2008). Our relationships and interpersonal connections are the most important part of our Quality World. We also have a 'perceived world' which represents the perception of our current reality. Our 'comparing place' exists between the two and is where we are constantly comparing what we want (quality world) with what we have (perceived world). If the two are out of balance, we can feel frustrated or distressed and start to behave in ways that will help us to get what we want. When the two worlds closely match, we feel good (The Glasser Institute for Choice Theory, 2019b). === Relationship habits === These habits represent the different ways in which we can choose to behave in response to the people around us. The 'connecting relationship habits' emphasise self-control. Practicing these healthy habits in our daily lives will increase our ability to make better choices and improve our relationships. In contrast, the 'disconnecting habits' represent external control. Engaging in disconnecting habits will ultimately lead to disconnection, resentment and the breaking down of our relationships (The Glasser Institute for Choice Theory, 2019b). '''''Table 1.''''' Side-by-side comparison of Choice Theory connecting and disconnecting relationship habits (The Glasser Institute for Choice Theory, 2019b). {| class="wikitable" !Connecting Relationships Habits !Disconnecting Habits |- |Supporting |Criticising |- |Encouraging |Blaming |- |Listening |Complaining |- |Accepting |Nagging |- |Trusting |Threatening |- |Respecting |Punishing |- |Negotiating Differences |Bribing, Rewarding to Control |} === Ten axioms of Choice Theory === * The only person whose behavior you can control is our own. * All we can give or get from other people is information. * All long-lasting psychological problems are relationship problems. * The problem relationship is always part of our present lives. * What happened in the past that was painful has a great deal to do with what we are today, but revisiting this painful past can contribute little or nothing to what we need to do now: improve an important, present relationship. * We are driven by five genetic needs: survival, love and belonging, power, freedom, and fun. * We can satisfy these needs only by satisfying a picture or pictures in our Quality Worlds. * All we can do from birth to death is behave. All behavior is Total Behavior and is made up of four inseparable components: acting, thinking, feeling and physiology. * All Total Behavior is designated by verbs, usually infinitives and gerunds, and named by the component that is most recognizable. * All Total Behavior is chosen, but we have direct control over only the acting and thinking components. Source: ''Choice Theory: A New Psychology of Personal Freedom'' by William Glasser, M.D. (as cited in The Glasser Institute for Choice Theory, 2019b). == What is reality therapy? == Reality Therapy is a method of counselling which utilises the principles of Choice Theory. Like Choice Theory, Reality Therapy states that behaviour is not caused by an external stimulus, but by what a person wants the most in the present moment. It was developed by Glasser in 1965 while he was working in a correctional institution. Reality Therapy can be seen as the practical application of choice theory and can be used therapeutically across a number of settings (Wubbolding et al. 2004). The primary objective of Reality Therapy is to assist people to connect with and move closer towards their Quality World and it functions within a number of key guidelines (The Glasser Institute for Choice Theory, 2019b): * Reality therapists have the job of helping people evaluate their current ability to achieve a particular want. *The therapist creates a supportive environment so that the client feels secure and can begin to make changes. *Clients are taught to accept responsibility and consequence for all behaviour and emotion, without excuse. *The focus is on what the client can do in the present. Clients are discouraged from dwelling on past events. *The client is encouraged to take action in the now and make better choices when they realise present behaviours are not getting them what they want. *Practical and efficient techniques, like goal setting are used to assist the client to change and track behaviour. *Individuals are assisted to evaluate their current situation and then to choose their own goals and follow through with them. * Responsibility and choice are the focus. Robert Wubbolding contributed to the teaching and practice of Reality Therapy through the development and implementation of various techniques and guidelines (Wubbolding, Casstevens, & Fulkerson, 2017). He developed the WDEP (wants, doing, evaluation, and planning) system of Reality Therapy which provides a practical framework which can facilitate person-centered planning: * Wants: "What do you want?" The client is asked to identify and prioritise their wants. * Doing: "What are you doing to get what you want?" The client is asked to identify the results of their current choices and where the choices are taking them. * Evaluation: "Is it working?" The client is asked to evaluate if their current choices are going to get them to where they really want to go. * Planning: "If not what can you do to meet your needs?" The client is assisted to create their own plan to fulfill their wants in an efficient and productive way that will not result in them hurting themselves or others. This model is designed to challenge the client and inspire self-evaluation. This will lead to the client making a plan which is designed to initiate positive change, which then leads to needs being met and connection in relationships bringing the person closer to their quality world. {{RoundBoxTop|theme=2}} ;Case study: The Use of Reality Therapy With a Depressed Deaf Adult Jerry was a 19-year-old male who lost his hearing at the age of four due to Otitis Media. He was self-referred for outpatient psychotherapy due to depression and identity issues negatively affecting his level of confidence and self-esteem. Jerry was seen by a clinician on 3 separate occasions. On the first occasion, the clinician implemented a WDEP method of therapy based around the following questions: "what do you want?, "what are you doing to get what you want?" "Is it working?" "If not what can you do to meet your needs?" Asking the client these questions helped him to investigate his quality world, and understand how to meet his own needs internally rather than externally. During session two, Jerry developed a plan as to how to achieve his treatment goals. In session three, he focused on the “doing” aspect of reality therapy and started implementing new choices, taking responsibility to change his own behaviour. After 12 sessions he ended up terminating the therapy, stating that he was no longer depressed, could identify confidently with the deaf world and planned to start pursuing a career as a certified horticulturalist (Bhargava, 2013). {{RoundBoxBottom}} == Do research findings support the application of Choice Theory? == The application of Choice Theory through Reality Therapy protocols has been well researched across a large number of settings. === School disciplinary problems === Class disruption, suspensions and detentions increase the amount of time students with behavioural issues are unsupervised and removing them from class may have an alienating effect on students, effecting{{gr}} their sense of belonging to a group or class (Walter, Lambie & Ngazimbi, 2008). Therefore according to Choice Theory, traditional disciplinary strategies may produce a negative cycle of student behaviour which results in further disengagement. Students who exhibit behavioral issues are at higher risk for depression, antisocial patterns in adulthood, gang involvement, and incarceration (Walter, Lambie & Ngazimbi, 2008). Implementing effective methods to address the behavioral problems of students early on is crucial. School-based Reality Therapy interventions teach students to develop a sense of personal responsibility and shift from an external to internal locus of control, allowing them to see how their chosen behaviours can directly affect their personal outcomes (Walter, Lambie & Ngazimbi, 2008). An example of this is in a study conducted in a middle school in the U.S.A. in which a protocol based on Choice Theory was implemented to a group of 6 students displaying ongoing behavioural issues (Walter, Lambie & Ngazimbi, 2008). The students were academically low achievers, who frequently blamed and criticised their teachers, peers and school for their behaviours. Based on the principles of Choice Theory, the school counsellor formed a "leadership group" from the students who were given the goal of making tangible improvements to the school over the course of 10 sessions. The purpose of this was to encourage the students to work together, forge connections and a support network and to enhance their individual social and emotional skills. The first 3 sessions focused on establishing trusting relationships between the students and counsellor and to establish individual roles within the group. The students decided as a group to organise a fun social event for the school, which would be their sole responsibility. The next five sessions were based around planning and implementing the event, with the counsellor's primary role to observe, support, and listen. In the final 3 group meetings the counsellor led the group in discussions where the students could share their thoughts about what the group did that worked, what they would do differently in future, how they felt about their individual contribution and what the project meant to them. During their time in the group, none of the students required disciplinary interventions and their teachers reported improvements to their overall behaviour and class engagement. This study showed that implementing a framework based on Choice Theory meant that the students could shift the ways in which they seek to satisfy their needs for belonging, fun and power and actively engage in a tasks in which they could practice making positive behavioural choices (Walter, Lambie & Ngazimbi, 2008). Some students with more extreme behavioural issues and less developed social skills may not be suitable for group work in this context. A case study written by a school counsellor looked at the effective use of a Choice Theory based therapy framework on a high-school student who was referred to them due to ongoing disruptive physical and verbal altercations with other students and teachers (Shillingford & Edwards, 2006). It was determined during two therapy sessions with the school counsellor that the student's negative behaviours were stemming from anger that he felt due to his father being incarcerated and bullying from his peers about the incarceration. Based on Choice Theory, the school counsellor hypothesised that the disconnection the student felt from his father and peers was the primary cause of his behavioural issues (Shillingford & Edwards, 2006). The school counsellor taught the student to recognise his needs and taught him that he was in control of his thoughts, behaviours and choices. The counsellor used modelling to represent positive habits and role-play to help the student practice implementing positive behaviours. He was also given homework tasks to help him practice choosing connecting relationship habits instead of disconnecting habits. This assisted the student to have more positive social interactions within the school which then led to enhanced feelings of connectedness. As a result of this therapy, the student's relationships with teachers and peers improved and he engaged in fewer altercations as the year progressed. === Addiction === Addictions tend to occur more frequently in individuals who have experienced emotional, physical or sexual abuse or have a dysfunctional family history. As a result of the trauma, these individuals may start using drugs or alcohol (for example) which eventually becomes integrated into their quality world (Mottern, 2002). Satisfying the addiction becomes a way to feel that their needs are being met. For example, drinking or using drugs with friends may fulfill the individual’s need for love and belonging or fun, and eventually relieving the addiction may fulfill the need for survival. Reality Therapy based treatments for addiction focus on teaching the individual to identify their own needs and start to make more appropriate choices to fulfill their needs and take responsibility for their own recovery. A 2014 study on female prison inmates used Reality Therapy to achieve better recovery from substance addiction (Law & Guo, 2014). They were split into a control group and an experimental group with 24 inmates in each group. Before the experiment, all participants were asked to complete a questionnaire designed to measure the inmate's sense of self-control. Questions included: "I am willing to take responsibility for making the effort toward the goals and plans I have set up for recovery," and "I believe I have the ability to refuse the temptation of drugs" among others. The participants in the experimental group then received 2-hour sessions of group-based Reality Therapy treatment once per week for a total of 12 weeks. The inmates in the control group did not receive any treatment, however were told they were on a waiting list for the same treatment program in another term. After the 12th session, the inmates completed the same questionnaire. When results were compared between the groups, the study found that the inmates in the experimental group significantly enhanced their sense of self-control and sense of self-determination. Enhancing self-control and self-determination may empower addicts to make healthier choices in line with their personal goals and learn to fulfill their needs more appropriately (Law & Guo, 2014). Similar studies have investigated Reality Therapy protocols for the treatment of other addictions. Kim (2008) investigated the efficacy of a Reality Therapy based group treatment for college students with internet addiction disorder. 25 participants (20 male and 5 female) were randomly assigned into two groups- a control group of 13 participants and an experimental group of 12 participants who received 2 sessions of Reality Therapy based group counselling per week for 5 weeks (10 sessions in total). A 40-item questionnaire was administered to both groups before and after the study to measure the level of internet addiction based on the participants self-reported internet use. The study found that the participants in the experimental group displayed a significantly lower internet addiction level and internet usage than the control group at the end of the 5 weeks. Most addiction studies have been consistent with these findings. Both of these studies show that enhancing client's awareness of their behaviours and needs and encouraging client responsibility to choose more effective behaviours may be useful as a treatment for addictions. They also show that Reality-Therapy based group work may be more powerful than individual therapies in treating addiction as the accountability towards the other members encourages clients to stick to their commitments. Building connections with others with a similar experience to their own may enhance an individual's feelings of love and belonging, meaning that they are less inclined to seek out their addiction to fulfill their need (Kim, 2008). Further, group Reality Therapy in addicts has been shown to reduce depression, anxiety and stress and increase self-esteem (Massah et al., 2015). === Relationship counselling === The WDEP framework can also be used to improve outcomes for couples seeking counselling. The breakdown of communcation{{sp}} is often cited as the most prevalent presenting problem for couples in therapy. The WDEP gives therapists a framework in which instead of merely working to improve communication skills, couples are encouraged to establish goals together which focus on making changes in the relationship based on replacing the disconnecting habits with connecting relationship habits (Mahaffey & Wubbolding, 2016). People may also seek to blame their partners for ongoing relationship issues, this blame as well as the desire to control others can derail our love relationships. Reality Therapy seeks to put the responsibility back on the individual to change what is within their control and encourages them to stop focusing on the faults of their partners, who we cannot change (Bradley, 2014). == Criticisms of Choice Theory == There are a few criticisms of Choice Theory. Perhaps one of the most controversial is the premise that our behaviours and relationships are the cause of all symptoms of mental illness. Choice Theory ignores the biological and environmental components of mental illness and instead blames our behaviour for all symptoms of mental illness. Secondly, Reality Therapy is focused entirely on the present and avoids any focus on a person's past. However, there may be value in visiting a patient's past experiences to give insight to determine what has led to the development of a mental disorder or what we might need to change. Not focusing on the past means that a patient may not be able to appropriately deal with past traumas. Lastly, Reality Therapy ignores the benefits of psychopharmacological interventions which have been shown to be useful in many cases to help reduce mental health symptoms. In Reality Therapy there is no diagnosis of mental health conditions as outlines in the DSM-V (APA, 2013) which could be detrimental for patients who are in need of appropriate therapies for example, anti-psychotics (Wubbolding et al., 2017). == Conclusion == Choice theory states that all human behaviour is a choice geared towards the fulfillment of our universal and genetically based human needs. Reality Therapy, which is based on choice theory, as well as WDEP techniques, give therapists a practical and usable system for developing goal specific treatment plans. Helping individuals to identify their goals and needs, examine their behaviours and evaluate the efficacy of those behaviours towards meeting their wants and needs gives individuals a framework in which they are called to action in the here and now to make choices which align better with their goals and will greatly enhance their life and their connections. Choice Theory, Reality Therapy and WDEP protocol can be utilised across a number of different settings to assist people to live happier and more fulfilling lives. == Quiz == <quiz display="simple"> {Reality therapy rejects the past and focuses on the present only |type="()"} - False + True {The primary goal of reality therapy is to connect or reconnect people with what they consider their quality world |type="()"} - False + True {The basis of Reality Therapy is choice theory |type="()"} - False + True </quiz> ==See also== *[https://www.youtube.com/watch?v=FckebmElMa8 Dr. Glasser Reality Therapy and Choice Theory] (YouTube) *[https://www.youtube.com/watch?v=6VAc_zECzfg Choice Theory: Ellen Gelinas TED Talk] (You Tube) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) ==References== {{Hanging indent|1= American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). Arlington, VA: Author. Bhargava, R. (2013). The Use of Reality Therapy With a Depressed Deaf Adult. https://doi.org/10.1177/1534650113496869 Bradley, E. L. (2014). Choice theory and reality therapy: an overview. International Journal of Choice Theory® and Reality Therapy, XXXIV(1), 6-13. Retrieved from: https://www.wglasserinternational.org/wpcontent/uploads/bskpdf.manager/18_IJCTRTFALL2014.PDF Kim, J.U. (2008). The effect of a R/T group counseling program on the internet addiction level and self-esteem of internet addiction university students. International Journal of Reality Therapy, XXVII(2), 4-12. Retrieved from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.520.6598&rep=rep1&type=pdf Law, F. M., & Guo, G. J. (2014). Who Is in Charge of Your Recovery ? The effectiveness of reality therapy for female drug offenders in taiwan. International Journal of Offender Therapy and Comparative Criminology, 58(6), 672-696. Retreived from: https://doi.org/10.1177/0306624X12474976 Mahaffey, B.A., Wubbolding, R. (2016). Applying reality therapy’s WDEP tenets to assist couples in creating new communication strategies. The Family Journal: Counseling and Therapy for Couples and Families, 24(1) 38-43. Retrieved from: https://psycnet.apa.org/record/2015-54150-005 Marlatt, L. (2014). The neuropsychology behind choice theory: five basic needs. International Journal of Choice Theory® and Reality Therapy, XXXIV(1), 6-13. Retrieved from: https://www.wglasserinternational.org/wpcontent/uploads/bskpdf.manager/18_IJCTRTFALL2014.PDF Massah, O., Farmani, F., Karimi, R., Karami, H., Hoseini, F., Farhoudian, A. (2015). Group reality therapy in addicts rehabilitation process to reduce depression,anxiety and stress. Iranian Rehabilitation Journal, 13(23), 42-48. Retrieved from: https://www.researchgate.net/publication/275232515_ Mottern, R. (2002). Using choice theory in coerced treatment for substance abuse. International Journal of Reality Therapy, XXII(1), 20-23. Retrieved from: https://www.academia.edu/7640414/Using_Choice_Theory_in_Coerced_Treatment_with_Substance_Abuse?auto=download Robey, P., Burdenski, T. K., Britzman, M., Crowell, J., & Cisse, G. S. (2011). Systemic applications of choice theory and reality therapy: an interview with glasser scholars. The Family Journal: Counseling and Therapy for Couples and Families, 19(4), 427–433. https://doi.org/10.1177/1066480711415038 Shillingford, M.A., Edwards, O.W. (2006). Professional school counselors using choice theory to meet the needs of children of prisoners. American School Counseling Association, 12(1), 62–66. Retrieved from: https://journals.sagepub.com/doi/abs/10.1177/2156759X0801200107 The Glasser Institute for Choice Therapy. (2019) About. Retrieved from https://wglasser.com/about/ The Glasser Institute for Choice Therapy. (2019) Quickstart guide to choice theory. Retrieved from https://wglasser.com/about/ Walter, S. M., Lambie, G. W., & Ngazimbi, E. E. (2008). A choice theory counseling group succeeds with middle school students who displayed disciplinary problems. Middle School Journal, 4–13. https://doi.org/10.1080/00940771.2008.11461666 Wubbolding, R. E., Brickell, J., Imhof, L., Kim, R. I., Lojk, L., & Al-rashidi, B. (2004). Reality therapy: a global perspective. International Journal for the Advancement of Counselling, 26(3), 219–229. https://doi.org/10.1023/B:ADCO.0000035526.02422.0d Wubbolding, R. E., Casstevens, W. J., Fulkerson, M. H., & Conceptualization, C. (2017). Using the WDEP System of Reality Therapy to Support Person-Centered Treatment Planning. Journal of Counseling & Development, 95, 472–477. https://doi.org/10.1002/jcad.12162 }} ==External links== *[https://wglasser.com/ The Glasser Institute for Choice Theory] [[Category:Motivation and emotion/Book/2019]] [[Category:Motivation and emotion/Book/Choice]] [[Category:Motivation and emotion/Book/Psychotherapy]] hd6o9r5ox0xc2wvxra63pt859cmj3n3 0 264911 2816083 2815395 2026-06-17T06:01:27Z Jtneill 10242 /* Google Scholar */ + link to journals 2816083 wikitext text/x-wiki {{Motivation and emotion/Tutorials|Tutorial 05: Functionalist theory and self-tracking|fifth}} {{Motivation and emotion/Tutorials/In development}} <!-- {{Motivation and emotion/Tutorials/Complete2}} --> <!-- {{Motivation and emotion/Tutorials/In development}} --> ==Overview== This tutorial: # explores practical applications of motivational theory using<!-- which can be used to understand why people behave the way they do - and how they can improve their goal striving and satisfaction / extend on content covered in Chapters 07 and 08 of [[Motivation and emotion/Readings/Textbooks/Reeve/2018|Reeve (2018)]] -->: ## '''functionalist theory of motivation''' with university student motivations as an example ## '''self-tracking''' as a form of feedback and self-monitoring # demonstrates use of '''Google Scholar''' to identify the top references on a topic ==Functionalist theory== # Models of motivation considered so far (e.g., models of needs) assume universality of motivation, and mostly don't recognise two key issues: ## People may perform the same behaviour but with different motivations ## There is often more than one motivation (reason) why someone performs a behaviour # The '''functionalist''' perspective of motivation (Clary & Snyder, 1999) proposes that: ## Behaviours serve different functions (or goals) for different people ## The match between a person's: (a) motivations and (b) outcomes determines (c) [[satisfaction]] and likelihood of continuing # A good match between motivations and outcomes leads to satisfaction and retention (or intention to continue), whereas if motivations are not well matched by corresponding outcomes this will lead to low satisfaction and heightened risk of drop-out # The take-home messages about motivation from a functional perspective are that: ## Motivations are multiple and complex (not singular and simple) ## Motivational profiles differ between people ## The match between motivations and outcomes predicts satisfaction which predicts likelihood to continue # It is also worth noting for further discussion that our motivational profiles fluctuate over time and between contexts # The following exercise demonstrates how a functionalist approach to motivation can be applied ==University student motivations== {{anchor|Activity}}{{RoundBoxTop|theme=10}} '''Activity: University student motivations''' # Consider: ##"'''Why''' are '''you''' at university?"—or, more generally, ## '''“Why do students go to uni?”''' # Develop a class [[w:Mind map|mind map]] of the main underlying motivations for attending university. Think and respond honestly—<u>''why''</u> are students <u>'''''really'''''</u> at university? ## Answers are likely to cover a wide range of human motives, but as the map develops, look for underlying themes and group similar motivations together. ## Past experience <!-- with this exercise --> and previous research finds that the motivations are likely to fall within these six categories: ##* '''Career/Qualifications''' - for the degree, so I can get a better job etc. ##* '''Self-Exploration/Learning''' - for the learning, curiousity, knowledge-seeking etc. ##* '''Social Opportunities''' - to meet people, make and explore friendships, enjoy social environment ##* '''Altruism''' - to become better able to help people, help society, help the planet etc. ##* '''Social Pressure''' - expectations of family, friends, society etc. ##* '''Rejection of Alternatives''' - better than doing nothing, working etc. <!-- (Note: Factor analytic research by Neill (2008) has not found psychometric support for the rejection of alternatives factor, but it has for the other five factors). --> # Complete the [https://docs.google.com/forms/d/e/1FAIpQLSdjfdQ7owYTDGTU4hNmBKRpyL_DZt4IilYZ4v-d12zNAKQiGA/viewform?usp=sf_link University Student Motivation] survey # Discuss results <!-- # Note and discuss: ## Where do you differ notably in your motivational profile from the university average? Who has a notable discrepancy that they would like to share? ##If a motivation factor was rated higher than its corresponding outcome, this is likely to contribute to dissatisfaction and risk of drop-out. ## If any outcome is rated higher than corresponding motivations, the experience of university is "over-delivering" in this area (i.e., you are getting more than expected) which may or may not contribute to satisfaction (depending on how valuable that outcome is to you). Include more info about Kerry Thomas' 4th year study with Lifeline and Red Cross Blood Donation study --> {{RoundBoxBottom}} ==Self-tracking== <div style="text-align:center">What are these objects?</div> <div style="overflow-x:scroll;overflow-y:hidden;"><div style="width:1560px;"><!-- -->[[File:Is this me mental disorder of looking.jpg|x70px]]<!-- -->[[File:Bullet-Journal-by-Matt-Ragland.jpg|x70px]]<!-- -->[[File:Pedometer.JPG|x70px]]<!-- -->[[File:Moodring2.jpg|x70px]]<!-- -->[[File:What's the Right Weight for My Height? (4254117120).jpg|x70px]]<!-- -->[[File:Blutdruckmessgeraet.jpg|x70px]]<!-- -->[[File:Elite HRV - Heart Rate Variability Reading.jpg|x70px]]<!-- -->[[File:Black Nike FuelBand.jpg|x70px]]<!-- -->[[File:Blood Glucose Testing.JPG|x70px]]<!-- --> </div></div> <div style="text-align:center">(mirror, bullet-journal, stop watch, mood ring, scales, blood pressure, heart rate variability, exercise activity tracker, blood glucose monitor<!-- , smart pill -->)<br>(they are self-tracking tools)<br><br>What are they for?<br>(learning about and improving ourselves)</div> # 21st century mobile applications offer an increasing array of self-monitoring tools. This presents an opportunity and a challenge: How can optimal use be made of self-tracking? # Discuss: ## What is self-tracking? ## What are some examples of self-tracking? What self-tracking do you do? What are you curious to try?<!-- # What have you discovered through self-tracking? What could you discover? --> # Watch: ## [https://www.youtube.com/embed/OrAo8oBBFIo?start=16&end=305 The quantified self] (Gary Wolf, TED@Cannes, 2010, YouTube, 4:49 mins) ## [https://www.youtube.com/watch?v=NP5okzCjrj0 The quantified self: Data gone wild?] (PBS NewsHour, 2013, YouTube, 5:45 mins) # Discuss: ## What are the potential benefits?<br>(e.g., self-awareness, steady stream of data-driven feedback) ## What are the potential problems?<br>(e.g., externalises motivation, could heighten sense of overwhelm and distress) ==Google Scholar== To help identify the best academic resources about a target topic, try these [http://scholar.google.com Google Scholar] search tips: <!-- # [http://scholar.google.com Google Scholar] e.g., [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=self-tracking+motivation self-tracking motivation] (search) --> # '''Citation rates''' - focus on sources with high citation rates (# of citations / years since published) # '''Authors''' - check the publications by top authors on the topic # '''Related articles''' - for top sources, check out "related articles" # '''Link to libraries''' - sync search results to UC Library holdings for quick access to restricted publications ## Login using Google Account ## Settings: Three bars (top-left) - Settings - Library links ## Search for institution name ("UC Library" is the main one, but also "University of Canberra" for Proquest) ## Select target libraries ## Save ## Search results will now show links to full-text resources held in the institutional library # [https://scholar.google.com/intl/en/scholar/help.html#library '''Storing citations'''] - save favourite publications to a folder. They can be found via My Library. # '''APA style citations''' - double-quote button. Provides a good start, but may need correcting, italics, and doesn't provide doi # [https://scholar.google.com/scholar_alerts '''Setting up alerts'''] - follow new publications about topics or authors of interest. "Create alert" at bottom of search. <!-- ## [https://canberra.libanswers.com/faq/192648 More info] --> <!-- # Filtering by year # '''Related articles''' # '''Citation searching''' (with Google Scholar or [http://ezproxy.canberra.edu.au/login?url=http://www.scopus.com/home.url Scopus]) --> <!-- # [https://scholar.google.com/intl/en/scholar/help.html#alerts Setting up alerts] --> # More info: ## [https://scholar.google.com/intl/en/scholar/about.html About] ## [https://scholar.google.com/intl/en/scholar/help.html Search tips] ## [https://guides.library.ucsc.edu/c.php?g=745384&p=5361954 Advanced search] # Other possibly useful search strategies: ## Include "review" or "meta-analysis" in the search to identify major reviews on the topic ## Search in [[Motivation and emotion/Journals|key journals]] ### Topic-specific - e.g., [https://link.springer.com/journal/11031 Motivation and Emotion] ### Major psychological review journals e.g., [https://www.annualreviews.org/journal/psych Annual Review of Psychology] ## [https://www.scopus.com/ Scopus] ## [https://scite.ai scite.ai] ==References== {{Hanging indent|1= Clary, E. G., & Snyder, M. (1991). A functional analysis of altruism and prosocial behavior: The case of volunteerism. In M. Clark (Ed.), ''Review of personality and social psychology: Vol 12. Pro-social behavior'' (pp. 119-148). Sage. Clary, E. G., Snyder, M., Ridge, R. D., Copeland, J., Stukas, A. A., Haugen, J., Miene, P. (1998). [http://www.comm.umn.edu/~akoerner/courses/5431-S13/Clary%20et%20al.%20(1998).pdf Understanding and assessing the motivations of volunteers: A functional approach]. ''Journal of Personality and Social Psychology'', ''74''(6), 516-530. https://doi.org/10.1037/0022-3514.74.6.1516 Clary, E. G., & Snyder, M. (1999). [http://cdp.sagepub.com/content/8/5/156.full.pdf The motivations to volunteer: Theoretical and practical considerations]. ''Current Directions in Psychological Science'', ''8''(5), 156-159. https://doi.org/10.1111/1467-8721.00037 }} ==Recording== * [https://au-lti.bbcollab.com/recording/b638e341c0404b7495a06ec2f4ee4889 Tutorial 05] (2025)<!-- * [https://au-lti.bbcollab.com/recording/09ecf4ffc3634700bbeebc40375ed75a Tutorial 05] (2024) * [https://au-lti.bbcollab.com/recording/80241f122b874ad48c34707e1362faa4 Tutorial 05] (2023) * [https://au-lti.bbcollab.com/recording/9c4925f4e1254be7bac2c1d37915e289 Tutorial 05] (2022) * [https://au-lti.bbcollab.com/recording/dd3e4ff54b6b427ba84d6f94285129ac Tutorial 05] (2021) --> ==See also== <!-- ;Additional tutorial material --> <!-- ;Book chapters --> ;Wikipedia * [[w:Biofeedback|Biofeedback]] * [[w:Quantified Self|Quantified Self]] * [[w:Uses and gratifications theory|Uses and gratifications theory]] ;Wikiversity * [[Feedback]] * [[Self-regulation]] * [[Volunteer motivation]] ;Lectures * [[{{#titleparts:{{PAGENAME}}|1}}/Lectures/Implicit motives and goals|Implicit motives and goals]] ;Tutorials * [[{{#titleparts:{{PAGENAME}}|2}}/Psychological needs|Psychological needs]] (Previous tutorial) * [[{{#titleparts:{{PAGENAME}}|2}}/Learned optimism|Learned optimism]] (Next tutorial) ;Admin * [[/Instructor notes/]] ==External links== * [https://www.google.com.au/search?q=bullet+journal+habit+tracker Bullet journals] (Google image search) * [https://www.abc.net.au/radionational/programs/sporty/fitness-trackers-and-cold-water-swimming/13512108 Do fitness trackers make you more active?] (ABC Radio National - Sporty podcast, 2021) * [https://www.youtube.com/watch?v=V08dWCtDyd8 The rise of the quantified self] (Mashable Brand X, 2014, YouTube, 1:48 mins) <!-- # An applied example: Workplace pedometer programs - http://www.10000steps.org.au, http://www.gettheworldmoving.com/ --> <!-- Find another video about pedometer based social media competitions --> <!-- ;Articles/Links # [https://www.youtube.com/watch?v=FESv2CgyJag The quantified self: How wearable sensors expand human potential] (Lauren Constantini, TEDxMileHigh, 2014) (10:55) # [http://www.technologyreview.com/biomedicine/37784/?mod=MagOur The measured life] # [http://www.forbes.com/sites/kashmirhill/2011/04/07/adventures-in-self-surveillance-aka-the-quantified-self-aka-extreme-navel-gazing/ Adventures in Self-Surveillance, aka The Quantified Self, aka Extreme Navel-Gazing] # [http://quantifiedself.com The quantified self] # [http://www.psychologytoday.com/blog/personal-science/201102/growth-quantified-self Growth of quantified self] # [http://opensource.com/health/11/8/open-health-quantified-self Open health with quantified self] # [http://www.launch.is/blog/bringing-quantified-self-to-the-masses-habit-labs-creates-ga.html Bringing quantified self to the masses: Habit labs creates games to make live healthier] # [http://www.fastcodesign.com/1665351/can-jawbone-really-made-people-healtheir-with-their-up-wristband Jawbone UP tracker] # [http://www.sbs.com.au/news/article/1641899/App-to-track-your-every-move App to track your every move] (PlaceMe) - [https://plus.google.com/111091089527727420853/posts/3iqjCACkBuz] # [http://news.cnet.com/8301-33620_3-57602925-278/how-my-body-rejected-activity-trackers-and-the-quantified-self/?google_editors_picks=true How my body rejected activity trackers and the 'quantified self'] (Danny Sullivan, 2013, CNET) --> <noinclude>{{Motivation and emotion/Tutorials/Navigation}}</noinclude> [[Category:Motivation and emotion/Tutorials/Functionalist theory and self-tracking]] pb8rajhbekxjhzjjjgkdwf862yp06lx 0 266105 2816028 2643307 2026-06-16T22:03:11Z Jtneill 10242 removed [[Category:Motivation and emotion/Book/Psychotherapy]] using [[Help:Gadget-HotCat|HotCat]] 2816028 wikitext text/x-wiki {{title|Cognitive behaviour therapy and emotion:<br>What effect can CBT have on emotion?}}{{MECR3|https://youtu.be/X2RABGjKgzk}} __TOC__ ==Overview== According to the Royal College of Psychiatrists in 2009 [[wikipedia:Cognitive behavioral therapy|Cognitive behavioural therapy]] (CBT) is one of the most effective treatments for conditions where anxiety or depression is the main problem as it can help patients break the vicious cycle of altered thinking, feeling and behaviour. CBT is the treatment of choice within the NHS (England based national health service) for depression and was developed by Beck in the early 1960s who found that it can be as effective as antidepressants for many types of depression{{fact}}. CBT can have positive effects on people's emotions by changing the way in which they perceive the events around them,{{gr}} it's framework mainly relies on participants developing a strong professional relationship with their psychologist to allow for fortnightly sessions that are based around the participant coming to terms with the things that are currently happening around them. This book chapter focuses mainly on the emotions of individuals who have mental health issues like depression and anxiety due to the amount of overall empirically accepted research on these issues compared to other mental health issues. {{RoundBoxTop|theme=2}} '''Focus questions:''' * Why is CBT important in emotions? * How can we examine the effects that CBT has? * What are the practical applications of this chapter? {{RoundBoxBottom}} == Cognitive behavioural theory == Cognitive behavioural theory focuses on changing unhelpful or unhealthy thoughts and behaviours. It is a combination of two therapies: ‘cognitive therapy’ and ‘behaviour therapy’. The basis of both these techniques is that healthy thoughts lead to healthy feelings and behaviours (Plate, Aldao 2017). [[wikipedia:: Cognitive therapy|Cognitive therapy]] is an intensive short-term psychotherapy which focuses on how patients make use of the information at hand in arriving at idiosyncratic interpretations and the effects of these views on their emotional experience. Many of the interventions in cognitive therapy set a goal of trying to “take a step back from” or to “distance themselves from” their thinking so that new information can be considered. Whereas [[wikipedia:Behaviour therapy|behavioural therapy]] aims to teach the person techniques or skills to alter their behaviour. [[Motivation and emotion/Textbook/Emotion/Theories/Cognitive|Cognitive theories of emotion]] have shown that there are a number of ways in which emotions can be influenced. CBT is important in emotions due to the nature of the therapy itself, by having a psychologist work closely with the participant it forces a conversation where reality has to be noticed by the individual. Beck realised the link between thoughts and feelings was important as he developed CBT,{{gr}} he invented the term 'automatic thoughts' which described emotion-filled thoughts that might pop up in the mind. In Beck's study he found that people weren't always fully aware of these thoughts and most of the time when negative thoughts occurred they weren't realistic or helpful. How CBT works: # First the automatic thought is identified # Then the validity of the automatic thought is questioned # Finally this challenges the core beliefs that the patient has <nowiki>In Freeman and Powers' 2007 review they stated that there have been numerous randomised clinical trials that support the efficacy and effectiveness of CBT for depression across a variety of clinical settings and populations (Clark, Beck & Alford 1999; De Rubeis & Crits-Christophe 1998; Dobson 1999; Robinson, Berman & Neimeyer 1990). {{</nowiki>[[Template:Missing|missing]]<nowiki>}} - New title; Criticisms of CBT. Critics of CBT argue that because the therapy only addresses current problems and focuses on very specific issues, it does not address the possible underlying causes of mental health conditions, such as an unhappy childhood (NHS Choices, 2010). Other criticisms of CBT include clients who have a problem with overintellectualising or those with minimal intelligence result in lower effectiveness. Weiner, Freedheim and Stricker in 2003 point out that “although the scope and efficacy of CBT are impressive, much work needs to be done. In particular, future efforts of CBT clinical researchers must demonstrate the effective of treatments outside research centres as well as turn more attention towards disorders overlooked by CBT (e.g., personality disorder)." Other criticisms lie in the strict framework in which CBT operates on which doesn't always allow for the resolution of underlying issues. From the psychologists' perspective CBT has been reported as a source of boredom and burnout as clients who refuse to face the reality around them from overintellectualising or other means end up repeating the same arguments with the therapist as the therapy loops in circles without any progress. Cognitive behavioural therapy (CBT) has been shown to be an effective treatment for depression and panic disorder in many randomized controlled trials (Gloaguen et al.,1998; Gould et al., 1995) However, more recent research has displayed that additional studies are needed to clarify the role of emotion regulation in CBT for panic disorder given the absence of such studies to date. However, findings from studies examining the role of emotion regulation in CBT for related anxiety disorders can serve as a basis for forming a hypothesis. For example, studies have shown that reappraisal increases in CBT for social anxiety disorder and that these increases generally predict subsequent reductions in anxiety symptoms, though findings regarding suppression are less conclusive (Goldin et al., 2014; Moscovitch et al., 2012).</nowiki> ==CBT and mental health issues== {{expand}} === Anxiety === Cognitive models of psychopathology tend to emphasise the role of dysfunctional thinking patterns in the development and maintenance of emotional disorders. [[wikipedia:Anxiety|Anxiety]] is a well researched mental health issue,{{gr}} the cognitive profile in the anxiety disorders is characterised by future-orientated automatic thoughts about potential physical or psychological threats or danger (Beck & Clark, 1988). Symptoms of anxiety disorders are usually divided into somatic and emotional symptoms. Somatic signs are manifested by abnormal bodily reactions such as rapid heart rate, muscle tension and excessive sweating. Successfully engaging in a pleasant activity may serve to enhance one’s expectancies for change. For example, anxiety and depression are often associated with poor social skill performance or performance on cognitive tasks {{gr}} due to this symptoms of anxiety disorder are often misinterpreted by more serious ailments (Shear, 2003). Treatment from the cognitive behavioural perspective assumes that anxiety is a normal, expected emotion comprised of biological, behavioural and psychological components. Individual risk for anxiety disorders vary given an individual’s genetic predisposition, temperament, family history, learning and environmental experiences, parenting styles, and other endogenous and exogenous factors. In its adaptive or ‘normal’ form, anxiety serves a protective function for the individual to alert him or her to danger and/or to motivate certain adaptive behaviours to avoid stress or negative experiences (Albano et al., 2009). CBT for childhood anxiety disorders has emerged as an efficacious psychosocial treatment approach (Albano et al., 2009). ===Depression=== Basic cognitive research has suggested that negative mood states are likely to lead to biases in attention, less efficient processing of information and poorer memory for mood-incongruent information, or perhaps selective memory for mood-congruent information. These findings are supported in clinical applications, impairments in attention, concentration and memory are frequent complaints of those suffering from anxiety disorders and depression. Clinical observations are full of examples of the ways in which reported events and their interpretations appear to be negatively biased. As in the lab (Bargh & Tota, 1988; Gotlib, McLachlan & Katz, 1988; Mogg et al., 1991), patients often report negative perspectives to their therapists in a fashion that is automatic and apparently without an awareness that there might be alternative point of view. Much of the work in CBT is designed to break the automaticity of negative thoughts and assumptions. [[wikipedia:Depression|Depression]] is another mental health issue that has been well researched. CBT is recommended by the Australia and New Zealand clinical practice when dealing with depression. Depressed individuals tend to be more self-focused or self-conscious than those who are not depressed (Ingram & Smith, 1984). Several theorists have argued that the tendency to self-focus, rather than engage in active coping may influence the course of depression (Ingram,1990; Lewinsohn et al., 1985). There has also been considerable research which demonstrates the difference of cognitive content between depression and the anxiety disorders when a tendency to self-focus is developed when the content of automatic thoughts is predominantly negative, stressing past losses and failures (Beck, 1967) depression is much more likely to occur. For example, a depressed individual tends to interpret personally negative experiences as evidence to support the view of oneself as a failure. Medications such as [[wikipedia:Antidepressant|antidepressants]] are effective for low/ mild cases of depression, when used in conjunction with CBT the rates of relapse or dependence are decreased. In severe depression it's recommended by the Royal College of Psychiatrists (2000) to use antidepressant medication in order to begin therapy after beginning to chemically change the way that the patient thinks. == Case study == Noluthando is a South African adolescent girl who was 17 when introduced to psychotherapy. She came from a problematic family home with an alcoholic mother and an abusive father to her mother. In Noluthando's mind she struggles to remember a time when neither of these problems existed. Noluthando was referred to a psychology clinic by a school counsellor after she had attempted suicide in late 2009. In a self-report, Noluthando felt that she was performing well at school and, while there were problems at home, she felt that she was coping in the earlier parts of 2009. Towards the end of 2009, Noluthando’s abusive father was diagnosed with HIV and this event reminded her of how her family doesn’t communicate effectively and the overall negative relationship between them. Noluthando had problems with displaying her emotions and set a goal of wanting to open up to people through CBT. There were communication issues from her early childhood, these issues made opening up for CBT difficult. As the therapy was slowly implemented, the therapeutic relationship flourished and became more collaborative. Importantly, this collaboration empowered Noluthando to take responsibility for her own therapy and improve her mood. Easterbrook and Meehan (2017), as the therapists from the study, commented on how CBT was more effective when the framework was not strictly followed but served rather as a guideline in which it was applied successfully. == Practical applications == [[File:Men Therapy Group - Scene from eMANcipation.jpg|thumb|413x413px|''Figure 1.'' Example of group men therapy.]] CBT has also been successfully used to treat many clinical conditions such as: * Gambling addiction * Substance abuse * Insomnia * Personality disorders * Psychosis * Stress management * Depression * Anxiety Cognitive approaches emphasise the role of thought in the development and maintenance of unhelpful or distressing patterns of emotion or behaviour. Treatments for anxiety and depression often begin with assignments for self-monitoring of critical events (Beck et al., 1979; Beck & Emery, 1985). Due to the flexibility of CBT it can be delivered in individual, group and couple formats. In practical applications of CBT monitoring can direct attention to the positive events that do occur which helps when avoiding recall biases often present in retrospective reports and can provide a more accurate/complete picture for therapist and client. Writing down one’s thoughts when experiencing negative emotions may help to slow down the process of thinking about and evaluating circumstances. Monitoring of variations in mood and cognition can also serve to reinforce the rationale and credibility of CBT by providing a first-hand demonstration of the correspondence between one’s mood and interpretation of events. This can lead to the breaking of the automatic habitual pattern of assumptions which lead to and contribute to maintaining negative mood states. Another technique which has practical applications from CBT is activity scheduling where a patient is encouraged to schedule and engage in simple time-limited activities which are enjoyable which helps facilitate efforts to consider alternative perspectives and encourage active coping. ==== How does CBT effect emotion practically? ==== The evidence supporting CBT informs us that as a theory it does have practical applications, the skills you learn in CBT are practical and helpful strategies that can be incorporated into everyday life even after the treatment is finished. Fundamentally the focus of CBT is to identify how problematic feelings are affecting their patients. CBT has the research behind it which has shown that it has the power to positively effect emotion by having solved a wide range of negatively charged emotional conditions like depression, anxiety and different types of addiction. == Quiz == <quiz display="simple"> {Which of the following mental health issues can be affected by cognitive behavioural therapy |type="()"} - Depression - Anxiety - Hypochondria + All of the above </quiz> ==Conclusion== CBT has many real-life applications and has been regarded as a theory supported by multiple counts of empirical evidence spanning over several decades. For future directions of research more research could on different psychological disorders like dissociative personality disorder, schizophrenia and even areas of psychosis still need to be further researched in order for CBT to be considered the standard at which all therapy should begin with. == See also == * [[Motivation and emotion/Book/2021/Cognitive evaluation theory and motivation|Cognitive evaluation theory and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/2019/Cognitive reappraisal of emotion|Cognitive reappraisal of emotion]] (Book chapter, 2019) * [[Motivation and emotion/Book/2017/Rational emotive behavior therapy|Rational emotive behavior therapy]] (Book chapter, 2017) * [[Motivation and emotion/Textbook/Emotion/Theories/Cognitive|Cognitive theories of emotion]] (Book chapter, 2010) ==References== <ref>{{Cite journal|last=Moscovitch|first=David A.|last2=Gavric|first2=Dubravka L.|last3=Senn|first3=Jessica M.|last4=Santesso|first4=Diane L.|last5=Miskovic|first5=Vladimir|last6=Schmidt|first6=Louis A.|last7=McCabe|first7=Randi E.|last8=Antony|first8=Martin M.|date=2012-08|title=Changes in Judgment Biases and Use of Emotion Regulation Strategies During Cognitive-Behavioral Therapy for Social Anxiety Disorder: Distinguishing Treatment Responders from Nonresponders|url=http://link.springer.com/10.1007/s10608-011-9371-1|journal=Cognitive Therapy and Research|language=en|volume=36|issue=4|pages=261–271|doi=10.1007/s10608-011-9371-1|issn=0147-5916}}</ref><ref>{{Cite journal|last=Goldin|first=Philippe R.|last2=Lee|first2=Ihno|last3=Ziv|first3=Michal|last4=Jazaieri|first4=Hooria|last5=Heimberg|first5=Richard G.|last6=Gross|first6=James J.|date=2014-05|title=Trajectories of change in emotion regulation and social anxiety during cognitive-behavioral therapy for social anxiety disorder|url=https://linkinghub.elsevier.com/retrieve/pii/S0005796714000266|journal=Behaviour Research and Therapy|language=en|volume=56|pages=7–15|doi=10.1016/j.brat.2014.02.005|pmc=PMC4136443|pmid=24632110}}</ref><ref>{{Cite journal|last=Forkmann|first=Thomas|last2=Scherer|first2=Anne|last3=Pawelzik|first3=Markus|last4=Mainz|first4=Verena|last5=Drueke|first5=Barbara|last6=Boecker|first6=Maren|last7=Gauggel|first7=Siegfried|date=2014-05|title=Does cognitive behavior therapy alter emotion regulation in inpatients with a depressive disorder?|url=http://www.dovepress.com/does-cognitive-behavior-therapy-alter-emotion-regulation-in-inpatients-peer-reviewed-article-PRBM|journal=Psychology Research and Behavior Management|language=en|pages=147|doi=10.2147/PRBM.S59421|issn=1179-1578|pmc=PMC4026562|pmid=24872725}}</ref><ref>{{Cite book|url=http://www.sciencedirect.com/science/article/pii/B9780128034576000052|title=The Science of Cognitive Behavioral Therapy|last=Plate|first=Andre J.|last2=Aldao|first2=Amelia|date=2017-01-01|publisher=Academic Press|isbn=978-0-12-803457-6|editor-last=Hofmann|editor-first=Stefan G.|location=San Diego|pages=107–127|language=en|doi=10.1016/b978-0-12-803457-6.00005-2|editor-last2=Asmundson|editor-first2=Gordon J. 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Cognitive behaviour therapy] (Better Health Vic) [https://www.counsellingconnection.com/index.php/2010/03/18/a-case-study-using-cbt/ Counselling case study using CBT] (Counselling connection) [https://www.heretohelp.bc.ca/visions/cbt-practice CBT in practice] (heretohelp) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Cognitive behavioural therapy]] 730vprdtjbtzs25iqrd8s1ranjs1ij7 2 266515 2815921 2815539 2026-06-16T12:35:47Z Alandmanson 1669821 /* Project: African Arthropods */ 2815921 wikitext text/x-wiki <!--Info--> https://www.archive.org {{list subpages|Alandmanson|User}} ===[[African Arthropods|Project: African Arthropods]]=== ;[[African Arthropods/Chelicerates|African Chelicerates]] :Arachnids and sea spiders — No sub-pages yet. ;[[African Arthropods/Crustaceans|African Crustaceans]] :Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods — No sub-pages yet. ;[[African Arthropods/Hexapods|African Hexapods]] :[[African Arthropods/Insects|African Insects]] :* '''[[African Arthropods/Diptera|Diptera]]''' :**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]] :* '''[[African Arthropods/Hymenoptera|Hymenoptera]]''' :**[[African Arthropods/Chalcidoidea|African Chalcidoidea]] :***[[African Arthropods/Eulophidae|African Eulophidae]] :***[[African Arthropods/Encyrtidae|African Encyrtidae]] :***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]] :***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]] :***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]] :**[[African Arthropods/Diaprioidea|African Diaprioidea]] :**[[African Arthropods/Platygastroidea|African Platygastroidea]] :**[[African Arthropods/Aculeata|African Aculeata]] :***[[African Arthropods/Apoidea]] :****[[African Arthropods/Crabroninae|African Crabroninae]] :****[[African Arthropods/Philanthus|South African species of Philanthus]] :***[[African Arthropods/Eumeninae|African potter wasps]] :* '''[[African Arthropods/Lepidoptera|Lepidoptera]]''' ;[[African Arthropods/Myriapods|African Myriapods]] :Centipedes, Millipedes, Pauropodans, Symphylans — No sub-pages yet.<br><br> ;Arthropods in South Africa :[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]] :[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']] :[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]] :[[African Arthropods/Pompilidae of South Africa|Pompilidae of South Africa]] ::[[African Arthropods/Pompilidae of SA with yellow wings tipped black|Pompilidae of SA with yellow wings, wingtips black]] ::[[African Arthropods/Pompilidae of SA with dark, blackish wings|Pompilidae of South Africa with dark, blackish wings]] <br> ===To Do=== Working on: [[User:Alandmanson/Hymenoptera of Africa]] Microgastrine cocoons in a net: <br> * http://www.waspweb.org/Chalcidoidea/Eupelmidae/Eupelminae/Eupelmus/Eupelmus/Eupelmus_species_2.htm * https://www.waspweb.org/Ichneumonoidea/Braconidae/Microgastrinae/Glyptapanteles/Glyptapanteles_acraeae.htm * https://commons.wikimedia.org/wiki/File:Microgastrinae_cocooncocoon_iNat_42943906.jpg * https://www.inaturalist.org/observations/38150348 * https://www.inaturalist.org/observations/144355729 * https://www.inaturalist.org/observations/39807090 * https://www.inaturalist.org/observations/145817446<br> [[Crop_production_in_KwaZulu-Natal|Project: Crop_production_in_KwaZulu-Natal]] [[Crop production in KwaZulu-Natal Annotated Bibliography]] [[Information for smallholders in KwaZulu-Natal]] [[Crop_production_in_KwaZulu-Natal/Climate-smart_Agriculture|Climate-smart Agriculture in KZN]] [[Plant propagation]]<br> <br> [[Animal Phyla/Arthropoda]]<br> [[:Category:Animals]]<br> [[:Category:Zoology]]<br> [[:Category:Entomology]] cbpyb5n3vliduvvajre3vw5j1w1cch2 2815922 2815921 2026-06-16T12:37:25Z Alandmanson 1669821 /* Project: African Arthropods */ 2815922 wikitext text/x-wiki <!--Info--> https://www.archive.org {{list subpages|Alandmanson|User}} ===[[African Arthropods|Project: African Arthropods]]=== ;[[African Arthropods/Chelicerates|African Chelicerates]] :Arachnids and sea spiders — No sub-pages yet. ;[[African Arthropods/Crustaceans|African Crustaceans]] :Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods — No sub-pages yet. ;[[African Arthropods/Hexapods|African Hexapods]] :[[African Arthropods/Insects|African Insects]] :* '''[[African Arthropods/Diptera|Diptera]]''' :**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]] :* '''[[African Arthropods/Hymenoptera|Hymenoptera]]''' :**[[African Arthropods/Chalcidoidea|African Chalcidoidea]] :***[[African Arthropods/Eulophidae|African Eulophidae]] :***[[African Arthropods/Encyrtidae|African Encyrtidae]] :***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]] :***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]] :***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]] :**[[African Arthropods/Diaprioidea|African Diaprioidea]] :**[[African Arthropods/Platygastroidea|African Platygastroidea]] :**[[African Arthropods/Aculeata|African Aculeata]] :***[[African Arthropods/Apoidea|African Apoidea]] :****[[African Arthropods/Crabroninae|African Crabroninae]] :****[[African Arthropods/Philanthus|South African species of Philanthus]] :***[[African Arthropods/Eumeninae|African potter wasps]] :* '''[[African Arthropods/Lepidoptera|Lepidoptera]]''' ;[[African Arthropods/Myriapods|African Myriapods]] :Centipedes, Millipedes, Pauropodans, Symphylans — No sub-pages yet.<br><br> ;Arthropods in South Africa :[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]] :[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']] :[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]] :[[African Arthropods/Pompilidae of South Africa|Pompilidae of South Africa]] ::[[African Arthropods/Pompilidae of SA with yellow wings tipped black|Pompilidae of SA with yellow wings, wingtips black]] ::[[African Arthropods/Pompilidae of SA with dark, blackish wings|Pompilidae of South Africa with dark, blackish wings]] <br> ===To Do=== Working on: [[User:Alandmanson/Hymenoptera of Africa]] Microgastrine cocoons in a net: <br> * http://www.waspweb.org/Chalcidoidea/Eupelmidae/Eupelminae/Eupelmus/Eupelmus/Eupelmus_species_2.htm * https://www.waspweb.org/Ichneumonoidea/Braconidae/Microgastrinae/Glyptapanteles/Glyptapanteles_acraeae.htm * https://commons.wikimedia.org/wiki/File:Microgastrinae_cocooncocoon_iNat_42943906.jpg * https://www.inaturalist.org/observations/38150348 * https://www.inaturalist.org/observations/144355729 * https://www.inaturalist.org/observations/39807090 * https://www.inaturalist.org/observations/145817446<br> [[Crop_production_in_KwaZulu-Natal|Project: Crop_production_in_KwaZulu-Natal]] [[Crop production in KwaZulu-Natal Annotated Bibliography]] [[Information for smallholders in KwaZulu-Natal]] [[Crop_production_in_KwaZulu-Natal/Climate-smart_Agriculture|Climate-smart Agriculture in KZN]] [[Plant propagation]]<br> <br> [[Animal Phyla/Arthropoda]]<br> [[:Category:Animals]]<br> [[:Category:Zoology]]<br> [[:Category:Entomology]] d4nmnfkw4yvbe8csbq6qnad22m7o0ma 2 266516 2815923 2787277 2026-06-16T12:43:44Z Alandmanson 1669821 2815923 wikitext text/x-wiki <!--Info--> |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] () |2={{clade |1={{clade |1=[[Heterogynaidae]] () |2={{clade |1={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} |2=[[Pompiloidea]] (spider wasps, velvet ants, and others) }} }} }} }} |2={{clade |1=[[Scolioidea]] (scoliid wasps) |2={{clade |1=[[Formicoidea]] (ants) |2='''Apoidea''' (spheciform wasps and bees) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 0vgdwua416zoz5is7g2glykyh1bguvu 2815924 2815923 2026-06-16T12:45:19Z Alandmanson 1669821 2815924 wikitext text/x-wiki <!--Info--> {{ |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] () |2={{clade |1={{clade |1=[[Heterogynaidae]] () |2={{clade |1={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} |2=[[Pompiloidea]] (spider wasps, velvet ants, and others) }} }} }} }} |2={{clade |1=[[Scolioidea]] (scoliid wasps) |2={{clade |1=[[Formicoidea]] (ants) |2='''Apoidea''' (spheciform wasps and bees) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 96b5dby4bxqgg61vl5nz0ud5v2bcabc 2815926 2815924 2026-06-16T12:58:10Z Alandmanson 1669821 2815926 wikitext text/x-wiki <!--Info--> {{ |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) |2={{clade |1={{clade |1={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} }} }} |2={{clade |1=[[Scolioidea]] (scoliid wasps) |2={{clade |1=[[Formicoidea]] (ants) |2='''Apoidea''' (spheciform wasps and bees) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == h9ikzm6ge9pk74xx4ghifdthcjnn695 2815929 2815926 2026-06-16T13:26:41Z Alandmanson 1669821 2815929 wikitext text/x-wiki <!--Info--> |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) |2={{clade |1={{clade |1={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} }} }} |2={{clade |1=[[Scolioidea]] (scoliid wasps) |2={{clade |1=[[Formicoidea]] (ants) |2='''Apoidea''' (spheciform wasps and bees) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == tiohy0qureswkblt7v2c1m0rghffc1t 2815962 2815929 2026-06-16T15:25:11Z Alandmanson 1669821 2815962 wikitext text/x-wiki <!--Info--> |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} }} }} |2={{clade |1=[[Scolioidea]] (scoliid wasps) |2={{clade |1=[[Formicoidea]] (ants) |2='''Apoidea''' (spheciform wasps and bees) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 8xkztwfp0lh5gi3y3rltfq7sbmk23q3 2815964 2815962 2026-06-16T15:32:51Z Alandmanson 1669821 2815964 wikitext text/x-wiki <!--Info--> |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} |2={{clade |1=[[Astatidae]] (scoliid wasps) |2={{clade |1=[[Formicoidea]] (ants) |2='''Anthophilaa''' (bees) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == sith6e3qka9xorhi6ii671qcmh1r15c 2815967 2815964 2026-06-16T15:50:45Z Alandmanson 1669821 2815967 wikitext text/x-wiki <!--Info--> |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} |2={{clade |1=[[Astatidae]] (scoliid wasps) |2={{clade |1={{clade |1=[[Pemphredonidae]] ( ) |2={{clade |1=[[Philanthidae]] |2={{clade |1=[[Eremiaspheciidae]] ( ) |2=[[Entomosericidae]] ( ) }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] ( ) }} }} }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == meb4ax7t93hw5x11hnl2lkt1lijma1a 2815968 2815967 2026-06-16T15:52:28Z Alandmanson 1669821 2815968 wikitext text/x-wiki <!--Info--> |cladogram={{clade|style=font-size:75%;line-height:75%; width:330px; |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} |2={{clade |1=[[Astatidae]] (scoliid wasps) |2={{clade |1={{clade |1=[[Pemphredonidae]] ( ) |2={{clade |1=[[Philanthidae]] |2={{clade |1=[[Eremiaspheciidae]] ( ) |2=[[Entomosericidae]] ( ) }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 7u92ifoc4im2abg3b1oqkd0ksw4qto7 2815969 2815968 2026-06-16T15:55:50Z Alandmanson 1669821 2815969 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} |2={{clade |1=[[Astatidae]] () |2={{clade |1={{clade |1=[[Pemphredonidae]] ( ) |2={{clade |1=[[Philanthidae]] |2={{clade |1=[[Eremiaspheciidae]] ( ) |2=[[Entomosericidae]] ( ) }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == l3z55ehxzv677qv92amq3syljp6lx64 2815970 2815969 2026-06-16T15:56:20Z Alandmanson 1669821 2815970 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} |2={{clade |1=[[Astatidae]] () |2={{clade |1={{clade |1=[[Pemphredonidae]] ( ) |2={{clade |1=[[Philanthidae]] |2={{clade |1=[[Eremiaspheciidae]] ( ) |2=[[Entomosericidae]] ( ) }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 7fm9mq6dshp5yk32yrrowsir6ys0sv7 2815971 2815970 2026-06-16T15:59:09Z Alandmanson 1669821 2815971 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] () |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] ( ) |2=[[Heterogynaidae]] ( ) }} |2={{clade |1=[[Sphecidae]] ( ) |2=[[Crabronidae]] ( ) }} }} |2={{clade |1=[[Astatidae]] () |2={{clade |1={{clade |1=[[Pemphredonidae]] ( ) |2={{clade |1=[[Philanthidae]] |2={{clade |1=[[Eremiaspheciidae]] ( ) |2=[[Entomosericidae]] ( ) }} }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == pbfuhld20u0zodmiwo9eod6rwnaqxqk 2815972 2815971 2026-06-16T16:25:07Z Alandmanson 1669821 2815972 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] () |2={{clade |1={{clade |1=[[Pemphredonidae]] ( ) |2={{clade |1=[[Philanthidae]] |2={{clade |1=[[Eremiaspheciidae]] ( ) |2=[[Entomosericidae]] ( ) }} }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == d5abnsabiyfu066igzi7poe46i643h6 2815973 2815972 2026-06-16T16:38:12Z Alandmanson 1669821 2815973 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] ( ) |2={{clade |1=[[Ammoplanidae]] |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 0mmo82u4i0uey51igxvz0wzvr0ff6x3 2815995 2815973 2026-06-16T18:39:24Z Alandmanson 1669821 2815995 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (bees) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 5dbxcn94x4nt41t2l5tpvwi4xpmvoll 2815996 2815995 2026-06-16T18:48:42Z Alandmanson 1669821 2815996 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 species in 99 genera; six families) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == 9knbetwrv1ru0b4l770lp5rysp9nnz2 2815997 2815996 2026-06-16T18:49:19Z Alandmanson 1669821 2815997 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families) }} }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == dz4i04ec8lnmly6wujokrmjwq0sum2c 2815998 2815997 2026-06-16T18:50:37Z Alandmanson 1669821 2815998 wikitext text/x-wiki <!--Info--> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families) }} }} }} }} }} }} }} = Pompilidae of South Africa = == South African Pompilidae with fore-wings mainly orange to yellow with fuscous (darker or blackish) wing-tips == <gallery mode=packed heights=200> Inaturalist 258649905 b.jpg Hemipepsis hilaris - inaturalist 10850475.jpg Cyphononyx decipiens inat 26259647 b.jpg Tachypompilus ignitus inaturalist 311015843 02.jpg Pompilidae 2021 12 12 inaturalist 313386858 04.jpg Pompilidae 2020 04 13 inaturalist 43563902 06.jpg </gallery> *The extent of the fuscous colour can be limited to the apex of the wing beyond the cells, or extend into the cells to a varying extent. * <br> == South African Pompilidae with fore-wings fuscous (black or very dark) == *The wings often have green-blue-violet reflections. <gallery mode=packed heights=200> Pompilidae 2019 05 01 2835.jpg|Female ''Batozonellus fuliginosus'' Pompilidae inaturalist 124148802 01.jpg|Female ''Cyphononyx optimus'' Pompilidae 2021 12 18 iNat 316501919 a.jpg|Female ''Cyphononyx obscurus'' Pompilidae 2025 03 14 iNat 266538336 a.jpg|Male ''Hemipepsis vindex'' Pompilidae_2019_05_28_0256.jpg| Spider-hunting Wasp (Hemipepsis) female (12640106905).jpg|''Hemipepsis'' sp. </gallery> <br> === Species with black antennae, legs, head, thorax and abdomen === Some parts may be brown. *''Java atropos'' *''Cyphononyx obscurus'' *''Hemipepsis vindex'' *''Hemipepsis vespertilio'' *''Hemipepsis braunsi'' *''Batozonellus fuliginosus'' <br> === Species with black antennae, head, thorax and abdomen, but legs (or parts of some legs) yellow to red === *''Cyphononyx optimus'' *''Paracyphononyx zonatus'' <br> <br> == South African Pompilidae with fore-wings mainly hyaline to fuscous-hyaline == <gallery mode=packed heights=200> Pompilidae inaturalist 123577538.jpg Pompilidae inaturalist 46961473.jpg Pompilidae iN 144781033 03.jpg </gallery> *With fuscous (darker) wing apex *One or two fuscous bands (faciated or bifaciated) *Hyaline parts can be clouded (whiteish clouding) or coloured (yellow-tinted) <br> == South African Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region: [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 Madl, 2020] *''Ceropales africana'' Móczar, 1989. - {{font color||yellow|''helvetica'' group}} (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales cribrata cribrata'' A. Costa, 1881; key in Móczár 1986a: 321 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales {{font color||#0f0|(Priesnerius)}} gessi'' Móczar, 1988 (South Africa) *''Ceropales {{font color||#0f0|(Priesnerius)}} grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales karooensis'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}} (Namibia, South Africa) *''Ceropales kriechbaumeri'' Magretti, 1884 - {{font color||yellow|''helvetica'' group}} (Burkina Faso, Nigeria, South Africa?, Uganda, Zimbabwe?) *''Ceropales {{font color||#0f0|(Priesnerius)}} kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Senegal, South Africa, Togo, Zimbabwe) *''Ceropales lawrencei'' Arnold, 1937 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales picta'' Shuckard, 1837; key in Móczár 1986b: 125 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus'' Cameron, 1904; key in Móczár 1986a: 320 (Lesotho, South Africa) **''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) **''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) **= Hemiceropales scobinifera (Arnold, 1937): Móczár 1986a: 319 *''Ceropales (Bifidoceropales) sulciscutis'' Cameron, 1910; key in Móczár 1990: 61 (South Africa, Tanzania) *''Ceropales waltoni'' Arnold, 1959 - {{font color||yellow|''helvetica'' group}}; key in Móczár 1989: 12 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) <br> ==Afrotropical Ceropalinae == Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region (Madl, 2020).<ref name=Madl2020>Madl, M. (2020). Annotated catalogue of the subfamily Ceropalinae (Hymenoptera: Pompilidae) of the Afrotropical region. Zeitschrift der Arbeitsgemeinschaft Österreichischer Entomologen 72: 73-84. [https://www.entomologie.at/permalink/articles/87-zeitschrift-der-arbeitsgemeinschaft-oesterreichischer-entomologen-72-2020-0073-0084 PDF]</ref> Ceropalinae can be defined by:<ref name=Brothers1993>Brothers, D. J. & Finnamore. (1993). Superfamily Vespoidea. In Goulet, H. & Huber, J. T. (Eds.). (1993). Hymenoptera of the world: an identification guide to families. 161-278. https://www.researchgate.net/publication/259227143</ref><ref name=Waichert2015> Waichert, C., Rodriguez, J., Wasbauer, M. S., Von Dohlen, C. D., & Pitts, J. P. (2015). Molecular phylogeny and systematics of spider wasps (Hymenoptera: Pompilidae): redefining subfamily boundaries and the origin of the family. Zoological Journal of the Linnean Society, 175(2), 271-287. {{doi|10.1111/zoj.12272}} [https://www.researchgate.net/publication/282015793 PDF]</ref> == Genera and species of Afrotropical Ceropalinae == This list is based on that of [https://www.waspweb.org/Pompiloidea/Pompilidae/Ceropalinae/index.htm '''waspweb'''] with changes following the Catalogue of Life (Kroupa & Schmid-Egger, 2025)<ref name=CoL2025> Kroupa, A. S., & Schmid-Egger, C. (2025). Hymenoptera Information System, Pompilidae of the World (version 2019-09). In O. Bánki, Y. Roskov, M. Döring, G. Ower, D. R. Hernández Robles, C. A. Plata Corredor, T. Stjernegaard Jeppesen, A. Örn, T. Pape, D. Hobern, S. Garnett, H. Little, R. E. DeWalt, J. Miller, T. Orrell, R. Aalbu, J. Abbott, C. Aedo, E. Aescht, et al., Catalogue of Life (Version 2025-07-10). Catalogue of Life Foundation, Amsterdam, Netherlands. https://doi.org/10.48580/dg9ld-4kv </ref> and [[w:George_Arnold_(entomologist)|papers by Arnold (1932-1962)]].<br> === Genus ''Ceropales'' === *''Ceropales africana'' Móczar, 1989. (Angola, Botswana, Burkina Faso, Central African Republic, Democratic Republic of Congo, Gabon, Gambia, Ghana, Ivory Coast, Kenya, Malawi, Namibia, Nigeria, Senegal, South Africa, Togo, Yemen, Zambia) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales arnoldi'' Móczar, 1988 (Namibia) *''Ceropales atra'' Móczar, 1991 (Botswana) *''Ceropales cribrata cribrata'' A. Costa, 1881 (Angola, Burkina Faso, Democratic Republic of Congo, Ivory Coast, Lesotho, Namibia, Nigeria, Russia, South Africa, Senegal, Tanzania, Togo, Zambia, Zimbabwe. Also Palaearctic region) *''Ceropales cribrata maculipes'' Móczar, 1986 (Zambia) *''Ceropales carinitifrons'' Wahis, 1986 (Madagascar) *''Ceropales angolaensis'' Móczar, 1989 (Angola) *''Ceropales dayi'' Móczar, 1989 (Kenya) *''Ceropales ferrugo'' Móczar, 1989 (Kenya) *''Ceropales gambiae'' Móczar, 1989 (Burkina Faso, Cameroon, Democratic Republic of Congo, Gambia, Nigeria, Senegal, Sierra Leone) *''Ceropales gessi'' Móczar, 1988 (South Africa) *''Ceropales grahamstowni'' Móczar, 1988 (South Africa, Zimbabwe) *''Ceropales juncoi'' Giner Mari, 1945 (Chad, Egypt, Israel, Pakistan, Somalia, Sudan, Western Sahara) *''Ceropales karooensis'' Arnold, 1937 (Namibia, South Africa) *''Ceropales kongoensis'' Móczar, 1988 (Burkina Faso, Democratic Republic of Congo, Ghana, Togo, Zimbabwe) *''Ceropales kriechbaumeri'' Magretti, 1884 (Burkina Faso, Nigeria, South Africa, Uganda, Zimbabwe) *''Ceropales latifasciatus'' Arnold, 1937 (Ethiopia) *''Ceropales lawrencei'' Arnold, 1937 (Botswana, Mozambique, South Africa, Zimbabwe) *''Ceropales levipleuris'' Wahis, 1987 (Madagascar) *''Ceropales maliensis'' Móczar, 1989 (Mali, Senegal) *''Ceropales maroccana'' Beaumont, 1947 (Burkina Faso, Democratic Republic of Congo, Gambia, Ghana, Ivory Coast, Nigeria, Senegal, Zimbabwe. Also Palaearctic region) *''Ceropales multipicta'' Arnold, 1937 (Botswana, Namibia) *''Ceropales picta'' Shuckard, 1837 (Democratic Republic of Congo, Ethiopia, South Africa, Uganda) *''Ceropales punctulatus punctulatus'' Cameron, 1904 (Lesotho, South Africa) *''Ceropales punctulatus bulawayoensis'' Bischoff, 1913 (Angola, Burkina Faso, Congo, Democratic Republic of Congo, Gambia, Ghana, Lesotho, Mali, Nigeria, Senegal, Sierra Leone, South Africa, Tanzania, Togo, Uganda, Zimbabwe) *''Ceropales punctulatus cereris'' Arnold, 1937 (Lesotho, South Africa) *''Ceropales ruficollis'' Cameron, 1910 (Kenya, Tanzania) *''Ceropales saegeri'' Móczar, 1988 (Democratic Reublic of Congo) *''Ceropales senegalensis'' Móczar, 1988 (Burkina Faso, Cameroon, Senegal) **''Ceropales senegalensis mbouri'' Móczar, 1988 (Senegal) *''Ceropales scobiniferus'' Arnold, 1937 (Democratic Republic of Congo, Mozambique, Nigeria, South Africa) *''Ceropales seyrigi'' Wahis, 1987 (Madagascar) *''Ceropales spinolai'' Móczar, 1988 (Guinea) *''Ceropales subhelvetica'' Móczar, 1988 (Burkina Faso, Senegal. Also Palaearctic: Israel) *''Ceropales sulciscutis'' Cameron, 1910 (South Africa, Tanzania) **''Ceropales sulciscutis raymondi'' Móczar, 1990 (Democratic Republic of Congo) *''Ceropales variolosus'' Arnold, 1937 (Democratic Republic of Congo, Ghana, Guinea, Mali, Nigeria, Senegal, Sudan, Togo, Uganda) *''Ceropales waltoni'' Arnold, 1959 (Botswana, Congo, Democratic Republic of Congo, Lesotho, South Africa, Zimbabwe) *''Ceropales yemeni'' Móczar, 1988 (Yemen. Also Palaearctic: Israel, Saudi Arabia) <br> === Genus ''Irenangelus'' === *''Irenangelus madescassus'' Wahis, 1988 (Madagascar) <br> ==Eumeninae== Photos of ''Antodynerus'' on GBIF:<br> ''alboniger'': https://www.gbif.org/occurrence/1248689053 (CC BY-NC-SA 3.0)<br> ''hova'': https://www.gbif.org/occurrence/1320165802 (CC0 1.0)<br> ''kelneri'': https://www.gbif.org/occurrence/3762658306 (CC BY-NC-SA 4.0)<br> ''lugubris'': https://www.gbif.org/occurrence/1248689125 (CC BY-NC-SA 3.0)<br> ''seyrigi'': https://www.gbif.org/occurrence/1322648015 (CC0 1.0)<br> ''sheffieldi'': https://www.gbif.org/occurrence/1318932924 (CC0 1.0)<br> ''silaos'': https://www.gbif.org/occurrence/1320574593 (CC0 1.0)<br> ==Ants== '''Subfamilies of Formicidae (WaspWeb)''' Number of iNaturalist records for subfamilies of Formicidae in Africa (2023-05-23) Amblyoponinae 7 Dolichoderinae 630 Dorylinae 1 167 Formicinae 10 396 Camponotus 6 090; Lepisiota 1 046 Myrmicinae 8 484 Crematogaster 1 786; Pheidole 1 468; Messor 1 156 Ponerinae 1 623 Proceratiinae 3 Pseudomyrmecinae 296 Aenictinae One Afrotropical genus ''Aenictus'' <br> Aenictogitoninae One Afrotropical genus ''Aenictogiton'' <br> Amblyoponinae Five Afrotropical genera <br> Apomyrminae One Afrotropical genus ''Apomyrma'' <br> Cerapachyinae Five Afrotropical genera<br> Dolichoderinae Eight Afrotropical genera<br> Dorylinae One Afrotropical genus ''Dorylus'' <br> Formicinae 20 Afrotropical genera<br> Leptanillinae One Afrotropical genus ''Leptanilla'' <br> Myrmicinae 37 Afrotropical genera <br> Ponerinae 18 Afrotropical genera <br> Proceratiinae Three Afrotropical genera <br> Pseudomyrmecinae One Afrotropical genus Tetraponera <br> <gallery mode=packed heights=200> Aenictogiton sp.jpg|''Aenictogiton'' sp., Aenictogitoninae Apomyrma stygia casent0101444 profile 1.jpg|''Apomyrma stygia'', Apomyrminae Cerapachys coxalis casent0173076 profile 1.jpg|''Cerapachys coxalis'', Cerapachyinae Cerapachys centurio castype12081-02 profile 1.jpg|''Cerapachys centurio'', Cerapachyinae Tapinoma subtile casent0132840 dorsal 1.jpg|''Tapinoma subtile'', Dolichoderinae Dorylus helvolus, a, Seringveld.jpg|''Dorylus helvolus'', Dorylinae Polyrhachis schistacea00.jpg|''Polyrhachis schistacea'', Formicinae Anoplolepis custodiens, met prooi, a, Krugersdorp.jpg|''Anoplolepis custodiens'', Formicinae AFRICAN THIEF ANT SIX.jpg|''Carebara vidua'', Myrmicinae Millipede Hunter Ant (Plectroctena mandibularis) (11904420373).jpg|''Plectroctena mandibularis'', Ponerinae Discothyrea hewitti sam-hym-c000061a profile 1.jpg|''Discothyrea hewitti'', Proceratiinae Probolomyrmex filiformis casent0102141 profile 1.jpg|''Probolomyrmex filiformis'', Proceratiinae Slender Ant (Tetraponera natalensis) (30538051244).jpg|''Tetraponera natalensis'', Pseudomyrmecinae </gallery> == N-P interactions == Dai, Z., Liu, G., Chen, H., Chen, C., Wang, J., Ai, S., Wei, D., Li, D., Ma, B., Tang, C., Brookes, P.C. and Xu, J., 2020. Long-term nutrient inputs shift soil microbial functional profiles of phosphorus cycling in diverse agroecosystems. The ISME journal, 14(3), pp.757-770. '''Abstract''' Microorganisms play an important role in soil phosphorus (P) cycling and regulation of P availability in agroecosystems. However, the responses of the functional and ecological traits of P-transformation microorganisms to long-term nutrient inputs are largely unknown. This study used metagenomics to investigate changes in the relative abundance of microbial P-transformation genes at four long-term experimental sites that received various inputs of N and P nutrients (up to 39 years). Long-term P input increased microbial P immobilization by decreasing the relative abundance of the P-starvation response gene (phoR) and increasing that of the low-affinity inorganic phosphate transporter gene (pit). This contrasts with previous findings that low-P conditions facilitate P immobilization in culturable microorganisms in short-term studies. In comparison, long-term nitrogen (N) input significantly decreased soil pH, and consequently decreased the relative abundances of total microbial P-solubilizing genes and the abundances of Actinobacteria, Gammaproteobacteria, and Alphaproteobacteria containing genes coding for alkaline phosphatase, and weakened the connection of relevant key genes. This challenges the concept that microbial P-solubilization capacity is mainly regulated by N:P stoichiometry. It is concluded that long-term N inputs decreased microbial P-solubilizing and mineralizing capacity while P inputs favored microbial immobilization via altering the microbial functional profiles, providing a novel insight into the regulation of P cycling in sustainable agroecosystems from a microbial perspective. ==Diptera== ===Wing and leg-waving behavior in flies=== ====Food detection==== *''Rhagio lineola'' and ''R. tringarius'' feed on pollen and/or honeydew, which they locate by sweeping their front legs across the surface of leaves. They have a few fine hairs on their front legs, probably for this purpose. Other Rhagionidae do not have these hairs. **https://www.researchgate.net/publication/359760392 *It is also possible that some flies sample the air with the chemical sensors on their legs or feet. **https://bugguide.net/node/view/217136/bgpage ====Courtship==== *Some Taeniapterinae are thought to wave their white-tipped front legs attract females. **https://bugguide.net/node/view/217136/bgpage *''Physiphora clausa'' appear to use leg-waving in courtship displays. **https://www.flickr.com/photos/jean_hort/4663220062 *Waving of forelegs is included in the complex courtship behavior of ''Physiphora demandata'' **https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1439-0310.1979.tb00298.x ====Mimics for defense==== *Stilt-legged flies ''Rainieria antennaepes'' mimic ichneumonid wasps. They extend their fore-legs in front of their head, so they look like wasp antennae. **https://thingsbiological.wordpress.com/2012/05/21/stilt-legged-flies-rainieria-antennaepes/ *Some hover-fly species mimic wasps by mock stinging, leg waving, or wing wagging. **https://www.jstor.org/stable/10.1086/674612 *Wing-waving to mimic salticid spiders. **https://www.researchgate.net/publication/27373081 https://www.researchgate.net/publication/6083895<br> <br> ===Number of iNat records in Acalyptrate fly families=== The [[w:acalyptratae|acalyptrate fly clade]] includes the following superfamilies and families:<br> * '''Carnoidea''' ** Acartophthalmidae 0 ** Australimyzidae 0 ** Braulidae (bee lice) 1 ** Canacidae (beach flies) 3 ** Carnidae (bird flies) 0 ** Chloropidae (frit flies) 259 ** Cryptochetidae 1 ** Inbiomyiidae 0 ** Milichiidae (freeloader flies) 158 <br> * '''Diopsoidea''' ** Diopsidae (stalk-eyed flies) 545 ** Gobryidae 0 ** Megamerinidae 0 ** Nothybidae 0 ** Psilidae (rust flies) 29 ** Somatiidae 0 ** Syringogastridae 0 <br> * '''Ephydroidea''' ** Camillidae 0 ** Campichoetidae 0 ** Curtonotidae (quasimodo flies) 15 ** Diastatidae 0 ** Drosophilidae (vinegar and fruit flies) 312 ** Ephydridae (shore flies) 117 <br> * '''Lauxanioidea''' ** Celyphidae (beetle flies) 0 ** Chamaemyiidae (aphid flies) 24 ** Cremifaniidae 0 ** Lauxaniidae (lauxaniid flies) 710 <br> * '''Nerioidea''' ** Cypselosomatidae 0 ** Fergusoninidae 0 ** Micropezidae (stilt-legged flies) 245 ** Neriidae 109 ** Strongylophthalmyiidae 0 ** Tanypezidae (stretched-foot flies) 0 <br> * '''Opomyzoidea''' ** Agromyzidae (leaf-miner flies) 161 ** Anthomyzidae 3 ** Asteiidae 4 ** Aulacigastridae 2 ** Clusiidae (druid flies) 2 ** Marginidae 0 ** Neminidae 0 ** Neurochaetidae 0 ** Odiniidae 0 ** Opomyzidae 4 ** Periscelididae 1 ** Teratomyzidae 0 ** Xenasteiidae 0 <br> * '''Sciomyzoidea''' ** Coelopidae (kelp flies) 51 ** Conopidae (thick-headed flies) 192 ** Dryomyzidae 1 ** Helcomyzidae 0 ** Helosciomyzidae 0 ** Heterocheilidae 0 ** Huttoninidae 0 ** Natalimyzidae 0 ** Phaeomyiidae 0 ** Ropalomeridae 1 ** Sciomyzidae (marsh flies) 67 ** Sepsidae (black scavenger flies) 269 <br> * '''Sphaeroceroidea''' ** Chyromyidae (golden flies) 19 ** Heleomyzidae (heleomyzid flies) 151 ** Nannodastiidae 0 ** Sphaeroceridae (lesser dung flies) 48 <br> * '''Tephritoidea''' ** Ctenostylidae 1 ** Lonchaeidae (lance flies) 47 ** Pallopteridae (flutter-wing flies) 5 ** Piophilidae (cheese skipper flies) 1 ** Platystomatidae (signal flies) 683 ** Pyrgotidae (scarab-pursuing flies) 119 ** Richardiidae 0 ** Tachiniscidae 2 ** Tephritidae (fruit flies) 1,759 ** Ulidiidae (picture-winged flies) 165 == References == nr2uinbe55ka210leo6libo3nc5wvwm 0 285380 2815941 2815802 2026-06-16T13:51:46Z Young1lim 21186 /* Applications */ 2815941 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.20260616.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> ogt9581bmx755pk8jibyy6m3dwka5pe 0 298877 2816027 2578202 2026-06-16T22:00:16Z Jtneill 10242 removed [[Category:Motivation and emotion/Book/Therapy]]; added [[Category:Motivation and emotion/Book/Psychotherapy]] using [[Help:Gadget-HotCat|HotCat]] 2816027 wikitext text/x-wiki {{title|Therapeutic recreation:<br>What is therapeutic recreation, how does it work, and what are its impacts?}} {{MECR3|1=https://youtu.be/M-LjjANVOK0}} __TOC__ ==Overview== {{RoundBoxTop|theme=4}} [[File:Recreational Therapist - DPLA - 6633d50d8e84ae373327d6878dc93da6.jpg|thumb|279x279px|'''Figure 1'''. A recreational therapist playing a game with a patient.]] Imagine you are at either work or school and today has been stressful, with numerous hassles and trivial tasks that have done nothing but bore you, making you yearn for a break from whichever routine you may be trapped in. These stressors get worse and worse until you cannot take it anymore and you need to find a way to deal with the day’s challenges, {{gr}} would you assume that recreational activities in between these stressors would allow you to do so? If so, what might these look like? {{RoundBoxBottom}} “The primary goal of human being is creating meaningful, healthy life of their existence” (Wise, 2021).” The concept of meaning in this context is defined as a process of discovering and achieving meaning from participating in beneficial activities (Morgan & Farsides, 2009). Due to one’s leisure being inherently unique and defined compared to other aspects of life (Walker, Kleiber, & Mannell, 2019), it holds potential to for individuals to gain value in their personal life (see Figure 1). Therapeutic recreation is designed to join recreation and medical treatment with the goal of improving clients’ wellbeing. This may be accomplished by improving the personal, spiritual, social or cultural aspects of the client’s life, each of which are key factors in what makes recreation meaningful altogether, thus the technic{{huh}} is versatile in nature, allowing for specialisation on one or more of these categories depending on the needs and wants of those involved (Armstrong & Manion, 2013). {{RoundBoxTop|theme=4}} ;Focus questions * What is therapeutic recreation? * How does therapeutic recreation work? * What are the impacts of therapeutic recreation? {{RoundBoxBottom}} == What is therapeutic recreation? == {{ic|Answer the question - what is therapeutic recreation?}} The multifaceted approach to patient care used in therapeutic recreation has been influenced by a range of psychological approaches over time. Originating from the medical model, it originally sought to "repair" patients by addressing perceived deficiencies or undesirable characteristics through personalised recreational activities aimed primarily at health enhancement (Mobily & Dieser, 2018; Austin & Van Puymbroeck, 2016). However, as psychological theories developed, therapeutic recreation has incorporated learnings from humanistic and social models, emphasising the emotional well-being of patients, the importance of leisure, and the social environment in which individuals live (Robertson & Long, 2020; Arai, Berbary & Dupuis, 2015). [[File:Dynamocamp-2017-17-gallery-1000x500.jpg|thumb|'''Figure 2''' An example of outdoor therapeutic recreation.]] === Medical model === The historical definition of therapeutic recreation focuses on the medical model, the presumption that something is “wrong” with the patient and thus require treatment to fix their illness in a personalised fashion as seen in Mobily & Dieser (2018) which focuses on repairing the individual (Mobily& Dieser, 2018). Austin and Van Puymbroeck (2016) provide another insight into this traditional model as the two conceptualise the treatment as a means of health maintenance through recreational activities which build upon a characteristic of the individual which is perceived as undesirable or unideal. This traditional model is the most common perspective of the treatment today and prioritises the health of the patient over the leisure which they may experience during said treatment (Austin & Van Puymbroeck, 2016). This initial view of therapeutic recreation shows that the treatment insinuates that for the patient to be involved with the treatment to begin with, one must be motivated to attain a different position post treatment that{{gr}} is considered desirable. === Humanistic approach === Although the medical model has been described as the most common and influential in the contemporary field by Genoe et al (2021), where there is one perspective, it is human nature for there to be at least one other (Genoe et al, 2021). Robertson and Long (2020) explore a more humanistic approach in ''Foundations of Therapeutic Recreation'' which acknowledges that for the treatment to be effective, it must appropriately utilise leisure and positive experiences to improve an individual’s health, partly through improved well-being and greater overall quality of life (Roberston & Long, 2020). This perspective builds upon the previously mentioned medical model by acknowledging that the primary focus is medical betterment, the emotional side of the client must be addressed and acknowledged appropriately, thus concluding that without the understanding of the patient’s emotions and the impacts which the treatment may have, the primary mission of the therapeutic practice will be undermined (see figure 2). === Social model === In recent years, different perspectives have risen following this acknowledgement of the patient’s emotional state by considering contemporary social justice. Arai, Berbary & Dupuis (2015) explore the social model of disability which focuses on changing the environment in which the patient finds themselves rather than directly treating their disability{{rewrite}}. To treat the disability insinuates that there is something wrong with the patient, which the social model argues is not always the case(Arai, Berbay & Dupui, 2015). Although more prominent in recent years, this emotion-based approach was first explored by Haun (1965). The milieu approach to therapeutic recreation proposed a universal viewpoint of individuals with inseparable genetic, personal, and real-life influences, key aspects of the person which must be considered for treatment. Haun (1965) acknowledged that a comfortable community within which the patient felt they belonged, as well as appropriate support from medical professionals when required, were essential to creating a pathway for treatment, proposing that this was not possible without positive interaction, and positive emotion consequently (Haun, 1965). Anderson (2021) further developed a personalised, emotional variant of the treatment by proposing an approach based on the strengths of the individual both internally and externally to further develop wellbeing for those affected by chronic conditions and disabilities in an individualised manner. This approach of course contradictory to the medical model, focuses on the emotional side of treatment and its ability to assist in personal development rather than “curing” a condition (Anderson, 2021). Choose your answers and click "Submit": {{RoundBoxTop|theme=6}} ;Quiz <quiz display="simple"> {The social aspect of therapeutic recreation has surfaced primarily in recent years: |type="()"} + True - False {Therapeutic recreation is, by definition, a "one size fits all" treatment. |type="()"} - True + False </quiz> {{RoundBoxBottom}} ==How does therapeutic recreation work?== Therapeutic recreation is a specialised approach designed to promote the well-being of individuals through engaging and meaningful leisure activities, offering interventions that are tailored to the strengths and needs of participants. It uses a strengths-based approach to create biopsychological changes that reinforce the positive outcomes of these interventions. === Strengths-based therapeutic recreation === Sylvester (1992) pioneered a modern strengths-based approach to this treatment which was designed to allow for the development of autonomy and confidence for patients by engaging in activities designed to establish and build upon faith in the self. To do so, the therapist would assume the role of a promoter of everyday autonomy and designing therapeutic activities which develop the client’s autonomy to conclude on a course of action and the autonomy to carry out said actions in the real world, thus developing decision making, typically for what many would consider trivial everyday decisions. These may range from deciding which recreational leisure’s{{sp}} to take part in, how to dress and which dishes to make and eat. Developing these decision-making skills, as well as providing an environment in which the patient can carry out their decisions, allows for the development of self-actualisation and the ability to engage in everyday practices, more importantly however, the ability to make these decisions with confidence (Sylvester, 1992). This was further developed by Anderson (2021) who built upon the strengths-based approach with their recreational practice referred to as “Flourishing through Leisure” which placed the patient, as well as their hopes, dreams, and aspirations, at the centre of the treatment process. Anderson (2021) propose{{gr}} a utilisation of the local environment and specialised therapeutic techniques to develop strengths in the form of emotional, cognitive, physical, social, and spiritual capabilities. The development of one or more of these strengths would allow the client to develop and therefore flourish. The situations in which these strengths develop are chosen by the participant and specialist, which allows for the specialist to create an individualised plan with enough versatility to adapt to the participant and the situations which they find themselves. This plan is also designed to produce leisure experiences for participant to allow for enjoyable pursuit of their goals and utilisation of whichever strength they are attempting to develop. While such development is occurring, the specialist encourages the participant to link different strengths and strength utilisations to different circumstances from various environment and contexts to better understand where and how these strengths may benefit the individual in everyday life. An example of this form of therapy is the utilisation of an adapted physical equipment in a safe environment to allow the patient to enhance their physical strength in a comfortable manner. Following the development of the patient’s strength, their self-efficacy and confidence should grow and allow them to engage in more meaningful leisure activities on their own accord, thus increasing their control over their lives and allowing for personal development (Anderson, 2021). === Biopsychological aspects === {{expand}} ==== Serotonin ==== [[File:Parasagittal MRI of human head in patient with benign familial macrocephaly prior to brain injury (ANIMATED).gif|thumb|<nowiki></nowiki> '''Figure 3''' An MRI showing every layer of the brain. ]] Models of therapeutic recreation are designed to exhibit desirable outcomes for the participants, however what exactly reinforces these outcomes and actions on a biological level is important to understand, to do so an investigation of neurotransmitters of the brain will be conducted (see figure 3). Matsunaga et al (2017) summarised serotonin as a neurotransmitter responsible for the regulation of mood, sleep and emotion and is found in higher quantities when an individual is engaging in activities which they find rewarding, challenging and for their own good. Serotonin imbalances can lead to mood imbalances and negative health outcomes, thus it is important to regulate the chemical to some extent, one method is of course achieving one’s potential – a key component of therapeutic recreation. Should the recreational therapy be fruitful for the participant and a challenge is faced, be it small or large in nature, serotonin will be regulated and flow, thus increasing their wellbeing (Matsungaga et al, 2017). ==== Dopamine ==== Serotonin does not work alone in the brain’s reward system however, it also works in conjunction with dopamine, the brain hormone responsible for the experience of joy, laughter and other positive emotions. As stated by Dfarhud, Malmir & Khanahmadi (2014), Dopamine can be released in a variety of circumstances, however the most relevant of which is fulfilling ones potential due to its nature of being tied to the reward system of the brain. On a neurological level, dopamine is produced in several areas of the brain including the substantia nigra, ventral tegmental area and the raphe nuclei, which also produces serotonin. Serotonin is also produced in enterochromaffin cells in the gastrointestinal tract which, although being far from the brain, still holds the ability to affect one’s mood and wellbeing (Dfarhud, Malmir & Khanahmadi, 2014). === Achievement motivation theory === Although the relation of biopsychological aspects of therapeutic recreation and its benefits has been acknowledged, it is worth questioning what psychological theories coincide with these effects and explain one another. Therapy of any kind is known to show little results unless the patient *desires* to improve their state or the state of the world in some form, therefore they are showing a motivation to achieve a goal. McClelland (2005) explores achievement motivation theory, which claims that motivations, as well as their ability to affect behaviour, consequently, varies between individuals, along with the contexts in which they encounter these situations, be them environmental or social. This theory claims that, although said circumstances may differ, humans by nature are motivated to accomplish tasks which they perceive as worthwhile (McClelland, 2005). This ties into the key principles of therapeutic recreation as the activities which are engaged in by patient and practitioner are designed to challenge and further the development of the patient in some way. These activities are of course difficult to accomplish in solitude, hence the patient engaging in this form of healthcare to begin with, therefore accomplishing these tasks will lead to productions of dopamine and serotonin, thus motivating the patient to continue with their therapeutic journey, living with a higher quality of life consequently. {{RoundBoxTop|theme=5}} ;Quiz <quiz display=simple> {Dopamine is responsible for : |type="()"} + Happiness and joy. - Doubt and fear. {Strengths-based therapeutic recreation focuses on: |type="()"} - Enhancing motivation. + Building self-efficacy and developing present talents. </quiz> {{RoundBoxBottom}} == What are the impacts of therapeutic recreation? == Therapeutic recreation is associated with improvements for various health conditions and emotional challenges. It can be effective in reducing symptoms of mood disorders like depression, assisting emotional regulation in those grieving, aiding recovery in chronic health conditions such as stroke and cardiovascular conditions, and contributing to the rehabilitation of cancer patients. Through diverse activities ranging from poetry to sports, therapeutic recreation offers a holistic approach to recovery and well-being.{{fact}} === Mood disorders: Depression === [[File:Depressed girl by brick wall.jpg|thumb|'''Figure 4''' Someone experiencing depressive symptoms. ]] Dieser and Ruddell (2002) tested the effects of therapeutic recreation on 10 adolescents aged 14-17 who were diagnosed with major depressive disorder and found a positive link between recreation, positive feelings, and alleviation from depressive symptoms (see Figure 4). Although the results indicate a relationship is present, the limited sample size hinders its generalisability. Thus, findings of Johnson (1999) should also be referred. Johnson (1999) discovered therapeutic recreation was useful in alleviating depressive symptoms amongst elderly participants, particularly due to its potential to be individualised. Poetry, dancing, and art were some of the forms which this therapy took, all of which were effective. An important finding was therapeutic recreation’s ability to aid in the treatment of physical conditions such as arthritis, as symptoms and complaints relating to these conditions became scarcer as treatment progressed, indicating a potential connection between the treatment and physical health as well as mental (Johnson, 1999). === Emotional regulation: Grief === Hanlon, Kiernan, and Guerin (2022) investigated the effects of therapeutic recreation on families who were struggling to engage in everyday activity following the death of one of their children from illness-related causes. 12 participants engaged in a three-camp program over a 12-month period, with a reunion camp 2 years later. The participants were encouraged to address their grief and allow for confrontation and acceptance of said grief with the goal of moving further past the loss of their child. Engagement with other families who lost a child, sharing experiences and understanding the gravity of their situations were the activities which the researchers encouraged. Results showed that participants unanimously felt the camp had aided in their grieving process, one of which told researchers the camp showed them ''“how to have fun again without forgetting [child] and others are going through the same”'' (Hanlon, Kiernan, and Guerin, 2022)''.'' Therapeutic recreation aiding in the recovery of loss was also found by King et al (2016) which saw participants who recently lost a family member engage in creative scrapbooking which involved poetry, mementos and photos. Participants reported the recreation allowed for an emotional outlet and allowed for grievances to be processed with less difficulty (King et al, 2016). === Chronic health conditions === {{expand}} ==== Stroke ==== Williams et al (2007) conducted a study on the effects of therapeutic recreation on adolescents and adults recovering from stroke in hospital, spending a total of 74.4% of their time in treatment. Recreational therapy was found to be a strong indicator of recovery from stroke, as well as increased functional independence and motor function, indicating that recreational therapy engagement may be a predictor of recovery from severe health conditions (Williams et al, 2007). Bode, Heinemann, Semik, & Mallinson (2004) reported hospitalised patients spent 5% of time in therapeutic recreation, which was then reported to have little effect on the patient's recovery from stroke. This further solidifies the relationship between therapeutic recreation and recovery from physical conditions such as stroke (Bode, Heinemann, Semik, & Mallinson, 2004). ==== Cardiovascular conditions ==== Allsop, Negly and Sibthorp (2013) conducted a study on the effects of therapeutic recreation on 79 teenagers with neurofibromatosis. Neurofibromatosis is a rare genetic disorder which sees tumours grow on the end of nerve cells, which leads to other issues surfacing such as cardiovascular issues, bone density inconsistency and learning disabilities (Noll et al., 2007). Those affected by this condition often struggle with social interaction, resulting in lower self-efficacy in social situations, which is the primary focus of the treatment. The researchers constructed programs that promoted social engagement, leadership, and independence. The control summer camp which saw social-focused therapeutic recreation therapy took place over several weeks, which saw greater social capability and self-efficacy compared to a group who did not engage with the treatment. Indicating the treatment holds potential to affect the lives of children facing chronic health conditions (Allsop, Negly and Sibthorp (2013). [[File:Hungarian U21 national soccer team (October 2019).jpg|thumb|'''Figure 5''' A team of soccer players.]] ==== Cancer ==== Uth et al (2013) also used therapeutic recreation to treat men with prostate cancer diagnoses. The men were engaged in soccer training multiple times per week and were encouraged to socialise with one another before and after practice over a 12-week period (see figure 5). Following treatment, participants were found to be more physically capable in terms of stair climbing and knee extensions, and were found to have higher lean body mass and increased physical health such as bone mass and increased plasma (Uth et al. 2013). Thus indicating therapeutic recreation has an impact on the rehabilitation and flourishment of cancer patients. ;Quiz <quiz display="simple"> {Therapeutic recreation studies show that the therapy leads to positive outcomes for those involved: |type="()"} + True - False {Social skills training is a form of therapeutic recreation, true or false? |type="()"} + True - False {Recreational therapy is useful for a variety of conditions that do not always gravitate toward mental illnesses. |type="()"} + True - False </quiz> {{RoundBoxTop|theme=2}} '''Case Study:''' Imagine you are a recreational therapist and a patient comes to you asking how they might be able to socialise better, as they have struggled with doing so their whole life due to ASD and believe they do not have the ability to speak to "normal" individuals. They also state they might give up and accept their status as a social outcast. What might you do to encourage healthy social interactions between this person and others? Perhaps basic social skills training to begin with? Or helping them build enough confidence to join a club of their interest? {{RoundBoxBottom}} == Conclusion == Therapeutic recreation is a versatile form of therapy which may range from treatments of emotional instability to chronic health conditions for individuals across the lifespan. This form of therapy is designed to be moulded depending on their {{who}} needs, desires and concerns in a way that typically promotes engagement, self-efficacy, health, and independence, allowing for an individual to both flourish in a medical sense, as well as the personal. Given the two are intertwined, it is of no surprise that therapeutic recreation builds upon one aspect of treatment to strengthen the other. Solo engagements between the patient and practitioner which may entail simple discourse to build social skills, or dancing to boost physical activity and capabilities, or group engagements that involve team sports such as soccer, or group discussions and interactions are some of the many forms which this therapy may take. A practitioner may weave the therapeutic technique to promote healthy brain chemistry also, {{gr}} perhaps to stimulate the development of dopamine to offset mood imbalances and combat chronic illnesses. Therapeutic recreation is not a form of treatment to be dismissed as impractical or ineffective, instead it is a medical doctrine which, like most therapies, is individualised, however more so due to its practical aspects and its design to produce desirable results which promote a healthy and enjoyable existence. ==See also== * [[Motivation and emotion/Book/2017/Serotonin and motivation|Serotonin and motivation]] (Book chapter, 2017) * [[Motivation and emotion/Book/2014/Depression and motivation|Depression and motivation]] (Book chapter, 2014) * [https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2016/Extreme_sport_motivation?wprov=srpw1_3 Motivation behind sport participation] (Book chapter, 2016) * [[wikipedia:Depression_(mood)|Depression article]] (Wikipedia) * [[wikipedia:Recreational_therapy|Another article on therapeutic recreation]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== {{Hanging indent|1= · Allsop, Negley, S., & Sibthorp, J. (2013). Assessing the Social Effect of Therapeutic Recreation Summer Camp for Adolescents With Chronic Illness. Therapeutic Recreation Journal, 47(1), 35–. · Anderson. (2021). Flourishing through leisure: An inclusive, ecological, and strengths-based approach. Loisir et Société, 44(2), 198–207. <nowiki>https://doi.org/10.1080/07053436.2021.1935411</nowiki> · Anderson, & Heyne, L. A. (2012). Flourishing through Leisure: An Ecological Extension of the Leisure and Well-Being Model in Therapeutic Recreation Strengths-Based Practice. Therapeutic Recreation Journal, 46(2), 129–. · Armstrong, L. L., & Manion, I.G. (2013). Meaningful youth engagement as a protective factor for youth suicidal ideation. Journal of Research on Adolescence, 25(1), 20–27. · Atkin. (1965). Recreation: A Medical Viewpoint. By Paul Haun. New York: Bureau of Publications, Teachers’ College, Columbia University. 1965. Pp. 98. British Journal of Psychiatry, 111(477), 786–786. <nowiki>https://doi.org/10.1192/bjp.111.477.786</nowiki> · Austin, D. R., & Van Puymbroeck, M. (2016). It is time for recreational therapists to declare themselves to be health care professionals. American Journal of Recreation Therapy, 15(1), 7–8. <nowiki>https://doi-org.ezproxy.canberra.edu.au/10.5055/ajrt.2016.0094</nowiki> · Arai, Berbary, L. A., & Dupuis, S. L. (2015). Dialogues for re-imagined praxis: using theory in practice to transform structural, ideological, and discursive “realities” with/in communities. Leisure = Loisir, 39(2), 299–321. <nowiki>https://doi.org/10.1080/14927713.2015.1086585</nowiki> · Bode, R., Heinemann, A., Semik, P., & Mallinson, T. (2004). Relative importance of rehabilitation therapy characteristics on functional outcomes for persons with stroke. Stroke, 55(11), 2537-42. · Dieser, & Ruddell, E. (2002). Effects of attribution retraining during therapeutic recreation on attributions and explanatory styles of adolescents with depression. Therapeutic Recreation Journal, 36(1), 35–47. · Dfarhud, Malmir, M., & Khanahmadi, M. (2014). Happiness & Health: The Biological Factors- Systematic Review Article. Iranian Journal of Public Health, 43(11), 1468–1477. · Genoe, Cripps, D., Park, K., Nelson, S., Ostryzniuk, L., & Boser, D. (2021). Meanings of therapeutic recreation: professionals’ perspectives. Leisure = Loisir, 45(1), 35–51. <nowiki>https://doi.org/10.1080/14927713.2021.1872411</nowiki> · Hanlon, Kiernan, G., & Guerin, S. (2022). Camp Draws You Back Into Life Again : Exploring the Impact of a Therapeutic Recreation-Based Bereavement Camp for Families Who Have Lost a Child to Serious Illness. Omega: Journal of Death and Dying, 302228221075282–302228221075282. <nowiki>https://doi.org/10.1177/00302228221075282</nowiki> · Johnson. (1999). Therapeutic Recreation Treats Depression in the Elderly. Home Health Care Services Quarterly, 18(2), 79–90. <nowiki>https://doi.org/10.1300/J027v18n02_05</nowiki> · King, Prout, B., Stuhl, A., & Nelson, R. (2016). Scrapbooking as an intervention to enhance coping in individuals experiencing grief and loss. Therapeutic Recreation Journal, 50(2), 181–. <nowiki>https://doi.org/10.18666/TRJ-2016-V50-I2-7308</nowiki> · Matsunaga, Ohtsubo, Y., Masuda, T., Noguchi, Y., Yamasue, H., & Ishii, K. (2022). Serotonin Receptor (HTR2A) Gene Polymorphism Modulates Social Sharing of Happiness in Both American and Japanese Adults. Japanese Psychological Research, 64(2), 181–192. <nowiki>https://doi.org/10.1111/jpr.12389</nowiki> · McClelland, D. (2005). Achievement motivation theory. Organizational behavior: Essential theories of motivation and leadership, 46-60. · Mobily, K., & Dieser, R. (2018). Seeking alternatives in therapeutic recreation/recreation therapy: A social/recreation community model. Leisure/Loisir, 42(1), 1–23. <nowiki>https://doi-org.ezproxy.canberra.edu.au/10.1080/14927713.2017.1403860</nowiki> · Morgan, J., & Farsides, T. (2009). Psychometric evaluation of the meaningful life measure. Journal of Happiness Studies, 10, 351–266. · Robertson, T., & Long, T. (2020). Considering therapeutic recreation as your profession. In T. Long & T. Robertson (Eds.), Foundations of therapeutic recreation (2nd ed., pp. 3–13). · Sylvester, C. (1992). Therapeutic recreation and the right to leisure. Therapeutic Recreation Journal, 26(1), 9–20. · Uth, Schmidt, J. F., Christensen, J. F., Hornstrup, T., Andersen, L. J., Hansen, P. R., Christensen, K. B., Andersen, L. L., Helge, E. W., Brasso, K., Rørth, M., Krustrup, P., & Midtgaard, J. (2013). Effects of recreational soccer in men with prostate cancer undergoing androgen deprivation therapy: study protocol for the “FC Prostate” randomized controlled trial. BMC Cancer, 13(1), 595–595. <nowiki>https://doi.org/10.1186/1471-2407-13-595</nowiki> · Walker, G. K., Kleiber, D. A., & Mannell, R. C. (2019). A social psychology of leisure (3rd ed.). Sagamore-Venture. · Williams, Barrett, J., Vercoe, H., Maahs-Fladung, C., Loy, D., & Skalko, T. (2007). Effects of Recreational Therapy on Functional Independence of People Recovering From Stroke. Therapeutic Recreation Journal, 41(4), 326–. Wise. (2021). Leisure and the Therapeutic Relationship: Contributing to Meaningful Lives. Therapeutic Recreation Journal, 55(2), 150–167. <nowiki>https://doi.org/10.18666/TRJ-2021-V55-I2-10739</nowiki> }} ==External links== * [https://www.dlgsc.wa.gov.au/sport-and-recreation/benefits-to-the-community Sports and recreation benefits] (Department of Local Government, Sport and Local Industries) * [https://www.qld.gov.au/recreation/health/get-active/benefits Benefits of being active] (Queensland Government) * [https://www.nctrc.org/about-ncrtc/about-recreational-therapy/ About therapeutic recreation] (National Council for Therapeutic Recreation Certfication) * [https://www.healthdirect.gov.au/dopamine#:~:text=Dopamine%20is%20responsible%20for%20allowing,of%20dopamine%20in%20the%20brain. Dopamine] (Health Direct) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Recreation]] [[Category:Motivation and emotion/Book/Psychotherapy]] bd5wv030tkly8pe4w5pmzv7m586ywur 0 301923 2815928 2783364 2026-06-16T13:12:50Z TMorata 860721 removed incorrect author names 2815928 wikitext text/x-wiki {{:Global Audiology/Header}}{{:Global Audiology/Asia/Header}}{{CountryHeader|File:Singapore on the globe (Southeast Asia centered).svg |https://en.wikipedia.org/wiki/Singapore}} {{HTitle|General Information}} [https://en.wikipedia.org/wiki/Singapore Singapore], officially the Republic of Singapore, is an island country and city-state in Southeast Asia. The country's territory comprises one main island, 63 satellite islands and islets, and one outlying islet. It borders the Strait of Malacca to the west, the Singapore Strait to the south along with the Riau Islands in Indonesia, the South China Sea to the east, and the Straits of Johor along with the State of Johor in Malaysia to the north. Singapore has four official languages: English, Malay, Mandarin, and Tamil. English is the common language, with exclusive use in numerous public services. {{HTitle|History of Audiology}} {{HTitle|Incidence and Prevalence of Hearing Loss}} {{HTitle|Information About Audiology}} {{HTitle|Scope of Practice and Licensing}} {{HTitle|Professional and Regulatory Bodies}} {{HTitle|Ongoing audiology research}} {{HTitle|Challenges, Opportunities and Notes}} {{HTitle|Audiology Charities}} {{HTitle|References}} <references responsive="" /> {{:Global Audiology/Authors-4|}} {{:Global Audiology/footer}} [[Category:Singapore]] [[Category:Audiology]] 7cowkjpirmiuqetftrwljm4kunerhqv 2 317234 2816033 2693768 2026-06-16T23:40:57Z Atcovi 276019 cat 2816033 wikitext text/x-wiki * [[User:Atcovi/Health Psychology/Chapter 1 - What is Health?]] * [[User:Atcovi/Health Psychology/Chapter 5 - Diverse Understandings of Stress]] * [[User:Atcovi/Health Psychology/Chapter 6 - Coping and Social Support]] * [[User:Atcovi/Health Psychology/Chapter 7 -Why Don’t We Do What We Need to?|User:Atcovi/Health Psychology/Chapter 7 - Why Don’t We Do What We Need to?]] * [[User:Atcovi/Health Psychology/Chapter 8 - Health Behaviors]] * SEPERATION * [[User:Atcovi/Health Psychology/Chapter 9 - Illness Cognitions, Adherence, and Patient–Practitioner Interactions: Introduction]] * [[User:Atcovi/Health Psychology/Chapter 10: Diverse Approaches to Pain]] * [[User:Atcovi/Health Psychology/Chapter 11: Disability, Terminal Illness, and Death]] * [[User:Atcovi/Health Psychology/Chapter 13: Cancer]] * [[User:Atcovi/Health Psychology/Chapter 14: Cardiovascular Disease]] [[Category:Atcovi/Health Psychology]] iq6iug0hkln6nbdq561arvpubqlcczj 2 317239 2816036 2692054 2026-06-16T23:42:10Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816036 wikitext text/x-wiki == 1.1 - What is Health? == === Introduction === * '''Health''' - state of complete physical, mental, and social well-being ([[w:WHO|WHO]]). ** How do we account for spirituality? ** Could we see health as a [[w:Teeter-totter|teeter-totter]], where optimal health is on one side while poor health is on the other (defined each by our habits)? But is it really as simple as excersising frequently, yet consuming a diet of mostly chips & soda (of course not!) Maintaining proper health is a daily commitment. === Diversity === * '''Intersectionality''' - Social + political impacts = Effect on health. Death rates are higher for black Americans vs. white Americans. * Answers about health can vary depending on '''culture''' (dynamic, but stable, set of goals, beliefs, and attitudes shared by a group of people, including sex, religion, and ethnicities) as well (between religions), financial state ($20k a year vs. $100k a year), and age (child vs. the elderly). * The '''Association of American Medical Colleges''' tries their best to alter their recommendations towards medical educators so they can address the health disparities amongst various cultures. === Cross-Cultural Views of Health === * '''Biomedical approach''' (commonly used in the Western world) focuses solely on the biological state of a human being (if there is no disease, then the individual is healthy!). * The '''Traditional Chinese Medicine (TCM)''' approach looks at health through the lens of the [[w:yin_and_yang|yin and yang philosophy]] (cold vs. hot qualities are balanced). * In Hindusim, the '''ayurveda''' accounts for health as “the ''three main biological units''—enzymes, tissues, and excretory functions—are in harmonious condition and when ''the mind and senses are cheerful''” (Agnihotri & Agnihotri, 2017, p. 31). * Some Mexican-Americans trust healers to cure spiritual problems, which is half of the problems that cause an illness (the other approach is essentially the biomedical approach: physical illness). * Native Americans look at a balance between human beings and the spiritual world (nature), and believe there are no distinctions between medicine and religion. == 1.2 - Defining Culture == === Introduction === * '''Culture''' can be defined as “a unique meaning and ''information system'', ''shared by a group'' and ''transmitted across generations'', that ''allows the group to meet basic needs of survival'', by ''coordinating social behavior'' to achieve a viable existence, to ''transmit successful social behaviors'', to pursue happiness and well-being, and ''to derive meaning from life''” (Matsumoto & Juang, 2017, p. 4). Can be split amongst several characteristics (Polish, a woman, rich, Jewish, for example). Culture cannot be just oversimplified as ethnicity or race. ** Culture, in fact, contains several different features, such as ethnicity, race, religion, being tall, geographical location, athleticism, computer science major, etc. It can be several components that make you, well—''you''. === Profile of a Multicultural American === * America is multicultural, full of multicultural individuals. ** Race: 72% White/Hispanics (Spain) or Latino (non-Spain), 13% Black, 6% Asian American, and 1% Native Americans. ** Religion: 71% Christian, 22.8% unaffiliated, 2% Jewish, .9% Muslims === Two Key Areas of Diversity === * '''Socioeconomic status (SES)''' - An entity(ies)'s economic and social level, measured by income, occupational status, and educational level - can also influence other factors, like race and the BMI index (latter in young adults). ** The more money you have (↑ SES), the healthier you are (through purchasing better foods, health insurance, medical services, education, etc.). * '''Sex''' - One's gender. ** Many health differences, such as life expectancies, are evident between men and women. ** Shaped by cultural expectations (see [[w:Drinking_culture_of_Korea#Dano]]), biological differences (estrogen and its protection against cardiovascular issues <50yrs), and sociographic expectations (social roles, such as women maintaining the development of the children while men work to provide for the families). === Advancing Cultural Competence === * [[w:Purnell_Model_for_Cultural_Competence|Purnell Model for Cultural Competence]] - 12 main cultural domains a clinician should explore with a client. == 1.3 - What is Health Psychology? == [[File:Bust of Hippocrates.jpg|thumb|The [[w:Hippocratic_Oath|Hippocratic Oath]] remains a big part of medicine to this day.]] === Introduction === * '''Health psychology''' is the area of psychology dedicated to the biological, psychological, and social factors behind promoting health and preventing illness. The subdivision of the APA dedicated to health psychology is known as the '''Society for Health Psychology'''. It is open to psychologists and other healthcare professionals interested in advancing the psychological aspects of mental/physical health. === The Evolution of Health Psychology === * Conceptualized as behavior medicine + medicine + array of public health sciences and services. * Emerged as a distinct field of study in North America in the 1960s. Came about after health professionals noticed humans were dying more from chronic diseases than other causes (such as famine). * ''Is the mind and body connected?'' Debated for centuries. Originally seen as one and 'spirits' were the cause of illness. Taoism & early Indian/Middle Eastern societies viewed them as connected. Ancient Greece challenged this notion and believed the mind and body were separate, as they valued rational thought. '''[[w:Hippocrates|Hippocrates]]''' believed in the happiness coming from the balance of four fluids. * '''[[Descartes]]''' comes with "I think, therefore I am", further strengthening the position of the Ancient Greeks - which allowed for us to study human anatomy more deeply. '''[[w:Galen|Galen]]''' first dwelled into animal dissection to find the causes of diseases. The study of human anatomy was "fine-tuned" by both [[w:Andreas_Vesalius|'''Andreas Vesalius''']] (1514–1564) and the Italian artist (and the prototypical Renaissance man) [[w:Leonardo_da_Vinci|'''Leonardo da Vinci''']] (1452–1519). Descartes had to mangle with the Roman Catholic Church to allow human dissections, as he reasoned that the mind and body were separate, and this is appearant at death (when the mind and soul leaves, and the body is left). Philosophy played a major role in the emergence of the health psychology field. * See [[AP Psychology/Introduction]] for a brief overview of the beginning of the field of psychology. * Psychoanalysts [[w:Franz_Alexander|'''Franz Alexander''']] and [[w:Helen_Flanders_Dunbar|'''Helen Flanders Dunbar''']] continued Sigmund Freud's work of attributing physical illness to psychological issues. They established '''[[w:Psychosomatic_medicine|psychosomatic medicine]],''' medicine dealing with the influence of minds on the health. Despite criticism for majorly holding onto Freud's beliefs (which are largely rejected/modified in today's day), the [[w:Society_for_Biopsychosocial_Science_and_Medicine|American Psychosomatic Society]] (APS) still exists to this day (founded in 1942). * '''[[w:Behavioral_medicine|Behavioral medicine]]''' examines non-biological influences on health, such as psychological issues. '''[[w:Society_of_Behavioral_Medicine|The Society of Behavioral Medicine]] (SBM)''' was founded in 1978. The [[w:Annals_of_Behavioral_Medicine|''Annals of Behavioral Medicine'']] is the journal of the SBM, akin to ''[[w:Psychosomatic_Medicine|Psychosomatic Medicine]]'' for the APS. * The '''[[w:International_Classification_of_Diseases|International Classification of Diseases]],''' which classifies diseases and disorders, is also a useful resource for psychologists. * Health psychology and medical sociology are influenced by '''[[w:Epidemiology|epidemiology]]''', a field of medicine which focuses on the "frequency, distribution, and causes of different diseases with an emphasis on the role of the physical and social environments". '''Morbidity''' is the number of cases of a disease that exist during a certain period of time, and '''mortality''' is the number of deaths related to a specific cause ([textbook citation TBD]). === Health Psychology’s Biopsychosocial Approach === * Our own biological makeup, culture, society, and/or our own thoughts, behaviors, and beliefs affects our behavior & health. This describes the '''biopsychosocial approach'''. The field of health psychology in a nut shell: # Stress and coping # Health behaviors # Issues in health care. [[Category:Atcovi/Health Psychology]] te3ky4t1cejp7r4af45442kw6ayv2g3 2 317255 2816037 2692403 2026-06-16T23:42:21Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816037 wikitext text/x-wiki == 5.1 - What is stress and how do we measure it? == '''Introduction''' * '''Stress''' - An upset of homeostasis (easiest definition). Measured as a stimulus, a response, and as an interaction. Stress can be subjective and different based on a person's personal experiences, but excessive stress leads to poor health outcomes and cause detrimental health consequences, such as heart attacks. * Majority of researchers agree the best way to know when a person is stressed is too look at how their ''body responds to a situation.'' Sympathetic nervous system activates? Their stressed! Stress in the early days were mostly biologically defined. ** '''Hans Selye''', in 1956, detected a specific response pattern which animals went through ([[w:General_Adaptation_Syndrome|General Adaptation Syndrome]]). ** Some saw stress as the "perceived demands on the organism that exceed the resources to meet those demands" (the demands are bigger than the resources). * A '''stressor''' is something that disrupts the body's 'homeostatic balance', while the '''stress response''' is the physical and mental response to the stressor. '''Measuring Stress''' * Two broad categories of stress measurement are '''physiological measures''' and '''self-reports'''. A '''self-report questionnaire''' (like the Life Experiences Survey or Social Readjustment Rating Scale [though widely criticized as it bases ''life events'' as 'automatic centers of stress', when that isn't always the case]) is a great way for someone to express if they are stressed or not. Other ways could be measuring the physiological response to stress, such as a blood test or measuring one's heart beat - but even this may not be accurate because other activities, not negative, stressful events, could trigger these physiological response. * The '''Hassles Scale''' consists of 117 events and accounts for 'small hassles', like a noisy neighbor, which can add up. Also found to be more accurate than the SRRS for measuring frequency and intensity of headaches/back pains in college students. * The '''Urban Hassles Index''' measures stressors that usually affect teenagers in urban environments. * Parents can benefit from the '''Parenting Daily Hassles Intensity Scale''' and the '''Family Daily Hassles Inventory'''. * Some other questionnaires shift from one-time events and daily hassles to ''major chronic stressors''. Examples are the '''Gurung 2004 21-item scale''' and the '''Trier Inventory of Chronic Stress'''. * The '''Perceived Stress Scale''' hones down on the subjectivity of stress, which differs person by person. One of the most commonly used stress scales in today's world. '''Stress Over Time''' * How are we stressed by certain things? Theorists like Walter Cannon and Robert Sapolsky believed that physiological responses to stress were developed through centuries of evolution. Early stressors were probably '''acute physical stressors''' in response to wild animals and beasts. Humans started to face '''chronic stressors''' as life expectancy increased. == 5.2 - Main Theories of Stress == '''Cannon's Fight-or-Flight Theory''' * Walter Cannon applied homeostasis to the study of human interactions with the environment. He looked into how stressors affect the '''sympathetic nervous system (SNS)'''. Activation of the SNS leads to increased circulation, respiration, and metabolism. Interestingly enough, the SNS turns off desires for food and sex. * When you are ''recovering'' from the activation of the sympathetic nervous system, your '''parasympathetic nervous system''' (PNS) takes activation. It does the opposite of the SNS by activating your digestion and reproductive system, and reduces your breathing and heart rate. * BOTH systems are coordinated by higher brain structures such as the [[w:Hypothalamus|hypothalamus]]. * Cannon argued that a stressor activates the SNS, which turns on the '''[[w:Adrenal_glands|adrenal glands]]''' (specifically the '''medulla''') that secrete '''[[w:Catecholamines|catecholamines]],''' with the major catecholamines being epinephrine (otherwise known as ''adrenaline'') and norepinephrine. This is known as the '''SAM activation'''. ''"An intricate dance of chemical secretions leads to all these events. The hypothalamus orchestrates the SNS via the secretion of corticotropin-releasing factor (CRF). CRF stimulates the secretion of adrenocorticotrophic hormone (ACTH) from the anterior pituitary gland and stimulates the locus coeruleus (located in the pons area of the brain stem) to increase the levels of norepinephrine in the system. Epinephrine is what increases both the heart rate and blood pressure. With prolonged stress, there is a circular reaction, and higher levels of epinephrine increase the secretions of ACTH. Research during the past 60 years has shown that the relative levels of epinephrine and norepinephrine vary with the type of emotion experienced with one being more of a flight chemical and the other being more of a fight chemical. Epinephrine is present in greater amounts when we are scared; norepinephrine is present in greater amounts when we are angry (Ax, 1953; Ward et al., 1983). The different physiological parts of SAM activation are heavily interconnected."'' * '''Freeze''' is also added to the "fight, flight" term. '''Taylor et. al.'s Tend-and-Befriend Theory''' * Women = Fight or flight + '''Tend-and-befriend''' by UCLA's Shelley Taylor and her colleagues. The argument is that in terms of evolution, women are not going to fight or run because they have to take care of their child - so they sought out support through connections/socialization. Womens' brains activate the attachment/caregiving side over the "crazy activation" of the fight-or-flight response. The female hormone, '''oxytocin''', counteracts the stress chemicals and improves bonding. '''Selye’s General Adaptation Syndrome''' * Selye came up with the notion that stressful events came about in a general, non-specific response to no matter the stressful events, driven by the '''hypothalamic-pituitary-adrenal (HPA) axis''' (limbic systemm + endocrine system that regulate the body's physiological reactions to stress through hormones). Found after his multiple rat experiments. ''"The first part of the HPA axis sequence of activation resembles the characteristics of SAM activation. The hypothalamus activates the pituitary gland that then activates the adrenal gland. '''The difference in Selye’s theory is that a different part of the adrenal gland, the cortex, gets activated'''. The cortex is the outer part of the adrenal gland (the medulla in SAM activation is the inner part) and '''secretes a class of hormones called corticosteroids'''. The major hormone in this class is cortisol (hydrocortisone). Cortisol generates energy to deal with the stressor by converting stored glycogen into glucose, a process called gluconeogenesis. Gluconeogenesis aids in breaking down protein, the mobilization of fat, and the stabilization of lysosomes."'' * Argued that organisms have a '''general adaptation syndrome''', a uniformed way where organisms respond to stress through alarm, resistance, than exhaustion (if stress is prolonged). Both scientists focused more on the physiological part of stress vs. the mental side. Neither scientist explained ''how'' someone recognizes something as threatening. This is where '''Richard Lazarus''' comes in (1966). '''Lazarus's Cognitive Appraisal Model''' * Stress = Imbalance between demands placed on individual and individual's resources to cope. Subjective, includes making/coming up with '''appraisals''', how a potentially stressful event is interpreted. We make two major types of appraisals... ** '''Primary appraisals''': Is the event positive, negative, or neutral? A '''harm appraisal''' is when we expect to lose or lose something of great personal significance. A '''threat appraisal''' is made when we believe the event will be demanding, harmful, and damaging. '''Challenge appraisals''' is similar to a growth mindset, where they view an event as a way to grow. Primary appraisals are influenced by our mastery or expectation of outcome. ** '''Secondary appraisals''': Can we deal with the event? If so, how? If no, then the event is a stressor. Cognitive appraisals can change the 'name of the game' in these situations. '''Stress, Hormones, and Genes''' * '''Receptor for Glucocorticoids''': Primary communicators of HPA axis stress responses in the brain and body. Similar with the '''serotonin transporter gene''' and the '''oxytocin receptor gene'''. == 5.3 - Factors Influencing Our Appraisals == * '''Duration of an even'''t: Acute vs. chronic. Staying over at your friend's for a week vs. 3 months. * '''Valence''': negative or positive. A negative experience when partying before would affect our appraisal for future partying. * '''Control''': Do we have control over the situation? The journey to being a professional soccer player, for example. * '''Predictability''': Do I know how long the doctor visit is during a busy day or not? * '''Definition''': Is it obvious or unknown? * '''Centrality''': Are we responsible for this? '''Culture and Appraisal''' * What would be a low-stress situation could be a high-stress situation elsewhere. For example, direct eye contact between Europeans is low-stress, while for Asian Americans, it is the opposite. * Conscious and nonconscious processes can contribute bias to an appraisal process. For example, if your family teaches you that requesting for ID is threatening in one way or the other, you'll have that fear regardless of the situation. * The '''cognitive appraisal model''' was expanded to include culture-specific dimensions. Occurrence of potentially stressful events can vary based on minority status, discrimination, or specific cultural customs. == 5.4 - Stress, Psychopathology, and Culture: The Diathesis-Stress Model == [[File:Diathese-stress-modell-1.svg|thumb|Diathesis-Stress Model (de)]] * '''Diathesis-Stress Model''': Relationship between ''vulnerable predispositions'', or diathesis, and stress as contributors towards psychopathological disorders. Stress activates the diathesis, which leads to psychopathology. '''Perceived Discrimination''' * If one perceives discrimination, it has horrible mental and physical health effects. Can be perpetrated by '''microaggressions''', small insults that are towards a marginalized group. * Discirmination can cause aging (shoter telomeres = increased aging). * '''Social isolation''' and '''low self-esteem''' were two sources of stress that originated from social discrimination. == 5.5 - Different Varietes of Stressors == * '''Relationships, work,''' and the '''environment'''. Family violence may be a more relevant stress compared to biological factors routinely checked during screening. '''Work Stress''' * '''Occupational stress''' has a place in the ''DSM-IV''. Could come from lack of knowledge of job knowledge, boredom, lack of control over activities, or having too much going on at once. * '''Stress contagion effect''': Stress from one sphere of life can affect another sphere of life (family vs. work spheres can affect each other). Findings explained by '''ecological theory''' (the different levels in which an individual acts in) + '''role theory''' (what is this individual's actions in the purpose in society that they fill?). ** '''Spillover''': Stress going from one sphere of life to another. ** '''Crossover''': Stress going from one individual to another. ** '''Ecological theory''': Consists of '''microsystems''', such as work and home domains, and '''mesosystem''', such as relationships and interactions between microsystems at a specific point in time. Due to '''reciprocity''', we know that systems are in interaction with each other. ** '''Role theory''': '''Role ambiguity''' is the level to which info regarding role expectations are readily available to the focal person. '''Role conflict''' arises when various roles come in the way of each other. '''Environmental Stress''' * Noise is a stress. Crowding as well. Divided into 3 main categories: # '''Background stressors''': Crowding and noise, including air/chemical pollution. # '''Natural disaster stressors''': Really bad short-term stressors, like [[w:Hurricane_Katrina|Hurricane Katrina]]. # '''Techno-political stressors''': 9/11, Boston Marathon bombing, or the [[w:Three_Mile_Island_accident|Three Mile Island accident]]. '''PTSD''' * Needs to exhibit symptoms >1 month after an event and decreased functioning. Includes migraine or bad respiratory health. * Found across several cultures and dealt with differently. Social support alleviates PTSD symptoms. == 5.6 - Consequences of Stress == * '''Allostasis''' - Ability to achieve stability through change. '''Allostatic load''' refers to the effects of chronic stress (restores the body’s homeostatic balance). Credit goes to '''Bruce McEwen''' for this. '''Allostatic Load''' (AL)''':''' "For most acute stressors, our sympathetic system is activated before and during the event (e.g., you have to make an oral presentation), and we adapt afterward. Even if this acute stressor is repeated a few times (e.g., you have to give a number of talks in a month), the healthy stress response shows an activation followed by a return to baseline functioning. In the case of chronic stress (e.g., living in a high-crime neighborhood where there are frequent stressors), AL is seen when the poststress adaptation or the normal lessening of the response for repeat stressors is not seen (b). You still respond, but it is a lower activation each time. Correspondingly, there is a prolonged exposure to the different stress hormones. This extra exposure can lead to a host of problems, such as coronary heart disease. Another result of AL takes place when our body is unable to shut off the stress response after the stressor stops. This again leads to extended exposure to stress hormones. The final case is system malfunctions in the response to stress. One system may not work, and other systems overcompensate. This also leads to extended exposure to stress hormones." * Markers of AL include hypertension, atherosclerosis, excessive fat storage, and poor sleep. Negatively effects memory. [[Category:Atcovi/Health Psychology]] 91lhxfvh6npnzffn98ovynl5luprha2 2 317312 2816038 2692533 2026-06-16T23:42:33Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816038 wikitext text/x-wiki ==6.1 - What Is Coping and How Do We Measure It?== * '''Coping''' - “constantly changing cognitive and behavioral efforts to manage specific external and/or internal demands that are appraised as taxing or exceeding the resources of the person” (Lazarus & Folkman, 1984, p. 141). Essentially, it's what we are doing to ''reestablish our homeostatic balances''. '''Social support''' is a critical component of coping and varies between cultures. Can be measured on an item level, such as drug use, to a "higher order level, mode, or style", as in investigating the coping methods employed by African Americans. * A ''coping style'' is different for everyone and may be different given the situations for one individual, this referring to ''situational coping''. Tons of influences go into your behavior and coping style when facing a stressor. * '''Moderators''' and '''mediators''' are the individual differences and different factors that influence the process. '''COPE''' * '''COPE Inventory''' - Psychological tool contrived to assess a broad range of coping responses. These subscales account for problem-focused and emotion-focused coping. Alcohol use was added later. '''Other Coping Questionnaires''' # '''Revised Way of Coping''' (RWOC) - 66 items, 4 point Likert-type response format. Written to allow comparisons across different types of stress situations, assessing the different ways of coping and measures of how each cope is used. # Other measures of coping are directed towards "special" populations, middle-aged to older-aged men, and specific coping behaviors. # Researchers do warn against the overuse of coping instruments, as these coping instruments ask too general of questions & neglect the unique aspects of different stressors, certain spects of coping behavior (such as the appropriateness of it), and the mix between coping style and/or personality dealing with a stressor. # The '''coping process''' is quantified and well-researched through a study of large samples. They also include interviews and observations, providing a ''mixed methods approach''. '''New Developments in Coping Research''' # Relationships are being studied more ('''relationship-focused coping''': maintaining relationships). Includes '''familism''' (stresses importance of family). # '''Daily process methodology''', or essentially "daily diaries", are a great way of assessing the "clear" process one has with coping. # New statistical techniques used to analyze complex data. == 6.2 - The Structure of Coping == * Two most basic styles of coping are '''approach-coping''', which is where you actively approach the problem at hand, and '''avoidant coping''', which is where you focus more on emotions than the actual stressor. * '''Coping strategies''' are the specific ways, either through behavior or mind, that people use to master or minimize stressors. ** '''Problem-focused coping''' involves dealing with the situation and working to resolve it. (ex, "How do I deal with an angry boss? Report them to HR!) ** '''Emotion-focused coping''' involves giving into "negative" emotions to ease the stress fo a situation. (ex, "I tell myself it's not real") *** Both focuses can be beneficial for a healthy human facing difficulties. **'''Avoidant coping''' is completely avoiding the situation, such as someone going to the movies to not think about their upcoming exam the next day. *'''B. F. Skinner''' & his colleagues believed that there are five categories of coping: support seeking, problem solving, avoidance, distraction, and positive cognitive restructuring. '''What is the best way to cope?''' * Generally, problem-solving coping > avoidant coping. * Some coping mechanisms may be both problem-solving and emotion-focused. * Best coping style depends on the '''severity, duration, controllability,''' and '''emotionality''' of the situation. ''The '''Psychosocial Aspects of Hereditary Cancer (PAHC)''' questionnaire is a 26-item questionnaire organized into six problem domains: genetics, practical issues, family, living with cancer, emotions, and children that screens for psychosocial problems and stress levels within the cancer genetic counseling setting.'' * Simple takeaway? Cope with the situation based on your comfort level. Avoidant/emotion-focused coping may be good for the short term to give you time to heal, then act. == 6.3 - Diverse Ways of Coping With Stress == Examining some of the factors that influence one's appraisals and coping: * One's '''personality''', or an individual's unique set of consistent behavioral traits, can be a set of "durable dispositions" to act in a certain way given a situation. For example, if you are 'easy-going', this trait of yours can help you cope with a situation. '''History of the [[w:Big_Five_personality_traits|Big Five]]:''' ''One of the earliest personality psychologists, Gordon Allport (1961), scoured an unabridged dictionary and collected more than 4,500 descriptors used to describe personality. Later personality theorists such as Cattell (1966) used statistical analyses to measure correlations between these different descriptors. Cattell found that all 4,500 descriptors could be encompassed by just 16 terms. It gets better. McCrae and Costa (1987) further narrowed these 16 terms down to a core of only five as part of their five-factor model of personality. A wealth of research has led to the Big Five or the Five Factor Model, suggesting that personality can be sufficiently measured by assessing how '''conscientious, agreeable, neurotic, open to experience, and extroverted''' a person is (John & Srivastava, 1999; Lee et al., 2005). And, personality traits can be the basis for individual differences in the biological response of stress (Soliemanifar et al., 2018). For an easy way to remember them, we like to use the acronym CANOE, or OCEAN if you live on the coast.'' * Big Five traits can influence health. "[[w:Type_A_and_Type_B_personality_theory#Type_A|Type A personalities]]" may lead an individual to '''Type A coronary-prone behavior pattern''', which is linked to coronary heart disease and stress. Being hostile is also a no-no for health, as hostile individuals may find support "stressful". * Coping styles ≠ reflections of personality. * Coping styles = mediate between personality & well-being. * '''Optimism, mastery, hardiness,''' and '''resilience''' all play roles in how we cope. Optimism are positive outcome expectations. Mastery is the regard that one has for control in their affairs and is also a mediator. Hardiness is the ability to take on and power through tough times, as these people see stress as something that makes life "more interesting". Resilience is withstanding difficult times. It can be studied under group settings and is affected by culture. ''Clauss-Ehlers (2008) found that cultural factors were related to measures of five aspects of resilience: childhood stressors, global coping, adaptive coping, maladaptive coping, and sociocultural support. Childhood stressors were experienced differentially by individuals from different racial/ethnic and social class status backgrounds, supporting proposals that ecological aspects, notably cultural background and experiences, influence the development of resilience. In another example, African American college students who received racial socialization messages (e.g., messages emp hasizing pride in being Black) and perceived that they had social support were more resilient (Brown, 2008). Similarly, Aboriginal Peoples in Canada have diverse notions of resilience grounded in culturally distinctive concepts of the person that connect people to community and the environment, the importance of collective history, the richness of Aboriginal languages and traditions, as well as individual and collective agency and activism (Kirmayer et al., 2011).'' == 6.4 - Culture, Coping, and Social Support == * East Asian cultures are '''collectivistic''', while US culture is '''individualistic'''. ''Specific coping methods within the collectivistic orientation included individualistic coping (coping alone through participation in solitary activities); seeking social support from family, members of own ethnic groups, or individuals who had gone through similar loss; forbearance (emotion-based coping); religiosity; and traditional healing practices. Coping strategies typically associated with individualistic cultures are approach-based, while avoidance-based coping strategies are often associated with collectivistic cultures (Chun et al., 2006).'' * '''Ethnic Identity:''' Strong ethnic identity is a strong factor to improved coping skills. Immigrants outside of their home have a significant disadvantage when looking at mental health. * '''Acculturation''': Important variable in stressors and negative health consequences, as non-European Americans are very much 'well aware' of their status as non-Whites, which can affect one's ethnic identity if surrounded by whites. Potentially serves as a mediator between perceptions of discrimination and distress, and varies between ethnic groups. ''Berry and colleagues (1986) define '''four models of acculturation''' that directly pertain to the issues we have raised here. A strong identification with both groups is indicative of integration or '''biculturalism'''; a strong identification with only the dominant culture reflects '''assimilation'''; with only the ethnic group, '''separation'''; with neither group, '''marginalization'''.'' '''Types of Social Support''' * '''Social support''' is always a positive factor in mental health, reduced mortality, and positive recovery from illness. ** Looking at the types of social support is important and is divided between "network" measures and "functional" measures. ** Functional support is broken down into '''received support''' (can be instrumental, informational, and emotional) and '''perceived support'''. '''Cultural Variables in Social Support''' * Despite Asian culture being largely collectivist, Asians are less likely to seek social support for coping with stress as the aim of keeping together is more important than potentially altering the 'harmony' between one another. * Gender also plays a role, with women having more strong relationships than men. * Latinas have stronger '''familialism''' vs. European Americans. '''Theoretical Perspectives on Social Support Change''' *'''Social Convoy model -''' People maintain their social circles as they grow older, despite their networks changing and possessing unique positives and negatives. Despite dropping some non-supportive folks, support will be either constant or increase. *'''Socioemotional selectivity theory''' - This theory proposes that as individuals perceive their time as more limited, they start to selectively reduce their social networks. They focus more on relationships that enhance their emotional well-being. Essentially, as people age, they prioritize spending time with those who contribute positively to their emotional state, leading to changes in social preferences over the lifespan == 6.5 - Keys to Coping With Stress == [[File:Aromatherapie.jpg|thumb|320x320px|Smell the scents ([[w:aromatherapy|aromatherapy]])!]] Two categories that help people cope better: * '''Relaxation-based approaches -''' Reduce cognitive load, activate PSNS, reduce sympathetic system. Includes meditation, yoga, guided imagery, and aromatherapy. Could use ''biofeedback'' (feedback of your biological [physiological] processes; also used in cognitive-behavioral therapy) to assist you in this process. ''Systematic desensitization'' is a form of classical conditioning where stressful thoughts/events are paired with relaxation. * '''Cognitive-behavioral approaches -''' Includes ''cognitive restructuring'' (fancy way of saying "change how one person thinks"), emotional expression (a diary), and exercise (influences metabolism of stress hormones). * Evidence of positive coping is feeling better! [[Category:Atcovi/Health Psychology]] fkz40rt7hqgiz28huw19tlx1o8wzy8e 2 317361 2816039 2692720 2026-06-16T23:42:42Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816039 wikitext text/x-wiki == 7.1 - What are Healthy Behaviors? == * '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). * '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology). '''The Healthy People Programs''' * The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats. '''What Determines Health Behaviors?''' * Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. ** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D₂ receptor gene is associated with alcoholism). ** '''Social factors''' include what/who you are exposed to and how these things can influence you. ** '''Psychological factors''' can include The Big Five personality traits. ''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).'' NOTE: Social & psychological factors can interact. == 7.2 - Changing Health Behaviors == * When setting a goal, bear in mind that you should set a goal through a "behavior contract" (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). ** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). ** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. '''Importance of Theory''' ''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.'' '''Key Theories of Health Behavior Change''' * Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most "central determinant" of health behavior change. In addition to SCT, we also have... ** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (<20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix "''Trans-''" is included because it identifies common themes across different intervention theories through the six separate stages. ** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM & culture can play a role as well. ** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture. ** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action. ** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out. Most predictive of behavioral intentions due to emphasis on self-efficacy. == 7.3 - Comparing the Models and Their Limitations == Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. * '''HBM''' lacks rigorously quantified data vs. the '''TPB''', but has recieved a lot of empirical support. For HBM, stuff like the beliefs about severity have low predictive value. * The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change. * The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness, but TPB DOES include congitive elements (attitudes, perceived beliefs, etc.). * The '''TTM''' does not allow individuals to skip steps, bringing criticisms that it is circular and flawed. '''Changing Behaviors: Interventions''' * '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. * What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''. [[Category:Atcovi/Health Psychology]] f4xz8k4dsbrl71iyjt3xac2h0nr8cms 2 317421 2816040 2692968 2026-06-16T23:43:00Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816040 wikitext text/x-wiki == 8.1 - Nutrition and Weight == * There is no single, specific, right answer for the nutritional standards for everyone. Everyone has a different metabolic and hormonal level. <u>Brief History</u> ''A public policy approach to making sense of the science of nutrition got started in the early 1900s when the U.S. Department of Agriculture (USDA) began to make recommendations to consumers about how much protein, fats, and carbohydrates to consume. Their first food guide was published in 1916 and consisted of five major food groups (e.g., fats, sugars; Welsh et al., 1993). The economic problems of the Great Depression in the 1930s greatly influenced American families’ food purchasing and consumption habits because they were forced to balance price and nutrition because affordable foods were often low in nutritional value. The rising inflation rates in 2021 and 2022 following the COVID-19 lockdown periods forced families to make similar decisions nearly 100 years later as food insecurity rates increased (NPR link 2022).'' * '''Food Guide Pyramid''' - A nutritional tool by the US Departement of Agriculture developed in the 80's - 2005, showing foods to consume in the bottom and foods to eat sparringly at the top. MyPyramid was released in 2005 to provide a sense of comprehensibility, something which the pyramid lacked. * In 2011, another guide was released, which is found on www.choosemyplate.gov. Changes included concise amounts of food consumption per age and gender, urging of more whole grains consumption + fruits&veggies consumption, and that 1/2 of the plate is fruits and veggies. ''Four factors were considered in establishing the serving sizes: typical portion sizes from food consumption surveys, ease of use, nutrient content, and traditional uses of foods (accounting for culture).'' **The Chinese culture includes "herbs of immortality" (bodybuilder type foods) and 'hot' and 'cold' foods, including spicy foods and expensive cuts of meats to vegetables, diary products, and inexpensive cuts of meat, respectively. **In Latin cultures, a "hot condition", like pregnancy, shouldn't be associated with "hot foods", which may bring about "hot illnesses", like skin rash. The opposite principle applies to the Chinese. **The "China Study" advocates for a vegetarian diet, but questions have been raised about its authenticity. *'''[[w:5_A_Day|5-A-Day program]]''' raises awareness regarding eating fruits and vegetables 5 times a day. '''Development of Food Preferences''' * Liking sweet and salty to sour is innate. Beyond this, preferences come into play. Behaviorist principles and culture plays a role as well. '''Obesity''' * Increase in obesity has occured between the 80s and now due to addition of [[w:high_fructose_corn_syrup|high fructose corn syrup]], change in play patterns for young children, and a rise in screen time. * '''Obesity''' - Having a BMI of 30>. Chance of obesity increases with age and is highest during mid-life. Commonly accepted cause is a combination of eating habits and lack of physical activity, but environment and genetics ([[w:ob/ob_mouse|''ob'' gene]]) can play a role too. Big cause is the supersizing fast food industry. * '''Overweight''' - Having a BMI of 25>. * BMI isn't entirely to be relied on, as it does not account for health risks, muscles, and cultural variations (difference in body fat distribution amongst various racial backgrounds). <u>Racial accounts in the US regarding obesity</u> ''The bottom line is that obesity varies significantly by different demographic groups (Hill et al., 2017; Wong et al., 2017). In a large national study, Black men have greater odds of obesity than White men in the South, West, and Midwest. In the South and West, Hispanic men also have greater odds of obesity than White men. In all regions, Asian men have lower odds of obesity than White men (Kelley et al., 2016). Further, Black and African American women are more likely to meet the criteria for obesity than either White women or Black men (Ogden et al., 2014; Wang & Beydoun, 2007). Recent work has also shown that transgender individuals experience higher risk of both underweight and overweight status, compared to their cisgender peers, at least in the college years (VanKim et al., 2014), while the risk for obesity is higher for transmasculine compared to transfeminine adults (Kyinn et al., 2021).'' * '''Sensory specific satiety''' - 1 food available, moderate amount; 2 foods available, 2nd food is eaten more than if it were to be presented by itself. * Presenting the food as 'healthier' or even 'cheaper' can change perceptions of that food. * Effects of weight discrimination and bullying due to weight also have severe impacts, including emotional distress, loneliness, and an increase in adhering metabolic syndromes and diabetes. '''[[Eating Disorders]]''' * '''Anorexia''' - Intense fear of gaining weight, characterized by excessive excersising, skipping meals, and indulging laxatives. * '''Bulimia''' - Binging on food, then purging. * '''Binge eating disorder (BED)''' - Binging food, but NOT purging. ''When compared with White American women, Black American women tend to have lower rates of anorexia nervosa but similar rates of BED, and Latinas may have slightly higher rates of eating disorders than both of these ethnic groups (Markey et al., 2009). Asian American women have the lowest rates of eating disorders among the major ethnic groups in the United States, while Native American women have the highest rates.'' == 8.2 - Physical Activity == * '''Physical activity''' is movement that is produced by contraction of the skeletal muscles, exerting energy. * Guidelines for physical activity include... ** Children & adolescenets (6-17yrs) should do 60m+ of activity daily and muscle strength work at least 3 days a week. ** Adults should engage in 150 minutes of moderate/75 minutes of high intensity excersise every week. Strengthening muscles should be done twice a week. * '''Basal metabolic rate''' - Number of calories you burn as your body performs basic (basal) life-sustaining function (https://www.garnethealth.org/news/basal-metabolic-rate-calculator) * '''Thermic effect of food''' - Amount of energy it takes for your body to digest, absorb, and metabolise the food you eat. * '''Exercise''' is physical activity with the goal of improving fitness. '''Cultural Variations in Physical Activity''' * The '''NHIS''', '''NHANES III''', and the '''Surgeon General's Report''', physical activity is lowest among people with low incomes and education. * Children with a TV in their bedroom are more likely to be less active. '''Health Compromising Behaviors''' * '''Behavioral cueing''' - A smoker always smokes on his work break, the work break becomes a cue to smoke. * '''Addiction''' - The body relies on substances over regular functioning. == 8.3 - Tobacco Use == * '''Tobacco use''' is the leading cause of preventable morbidity and mortality in the US. * Highest in ages 25 - 65, lowest in early adulthood. Memory recall and exposure increases risk of smoking. '''Cultural Variations''' * Men smoke more than women. * Less education & income, more likely to smoke. * KT & VA have the most smokers. * Native Americans, Alaskans, Blacks = Whites, Hispanic, Asian Americans (highest rate of smoking to least rate of smoking). '''Why do people smoke?''' * ''Why do people start'' and ''why do they keep smoking?'' ** Nicotine is pleasing, '''biologically'''. ** '''Genetics''': certain genotypes, like SLC6A3, DRD2, and DRD2-A1, have an association with smoking. ''Genes also influence how nicotine is broken down. Tang et al. (2012) investigated how nicotine metabolism and genetic variation in CYP2A6, a gene that mediates nicotine breakdown, influence the neural response to smoking cues. Tang et al. used functional magnetic resonance imaging to scan smokers with variations in the gene and hence high and low in nicotine metabolism. Fast metabolizers, by phenotype or genotype, had significantly greater responses to visual cigarette cues than slow metabolizers in the amygdala, hippocampus, striatum, insula, and cingulate cortex. This finding helps explain why fast metabolizers who smoke have lower cessation rates. Similar brain mapping of smokers with different genes show how variations in the amygdala (responsible for emotion and pleasure) may influence cessation (Jasinska et al., 2012).'' * '''Psychological''': Parental/peer modeling. Overcoming inferiority and establishing an identity may lead to smoking (Erikson). * '''Social/cultural''': Movies/ads. * '''Physiological''': Chances of dying increases. Has a '''synergistic effect''' (cumulative effects of two active ingredients). '''ETS,''' or ''environmental tobacco smoke'', is the tobacco smoke inhaled by non-smokers who are near a person who smokes. Passive smoking is linked to lung cancer, cardiovascular disease, and smoking addiction. == 8.4 - Alcohol == * 3rd leading cause of death. '''Who Drinks, How Much, and Why?''' * '''Alcohol abuse''' - Diagnosed either through 1) lack of fulfilling responsibilities, 2) using alcohol repeatedly whilst putting others in danger, 3) getting in trouble with the law due to alcohol on a regular basis, and 4) using alcohol despite obvious social problems. ** Men who consume ''five or more'' drinks in a row, women who have consumed ''four or more'' drinks in a row at least once in the last 2 weeks. * ''Why?'' ** '''Biologically:''' Genetic predictors of alcoholism. Reduces stress. ** '''Psychologically''': High in neuroticism, impulsive, and extroverted. If someone were to 'expect good things' from alcohol, then they are more likely to continue indulging in alcohol. Social peers influence alcohol consumption as well. '''Consequences of Alcohol Abuse''' * 7th leading cause of death worldwide, leading cause of early mortality between ages 15 - 45. * Liver disease is a common consequence for older drinkers. '''Benefits?''' * Recent studies have put out a '''standard drink''', like a 12-ounce serving of beer, as a standard serving for alcohol. Also, a citing of the '''[[w:French_paradox|French paradox]].''' Though recent evidence have shown that these studies showing the so-called "positive" effects of alcohol don't account for other healthy factors. The CDC still recommends non-drinkers to not drink alcohol. [[Category:Atcovi/Health Psychology]] kyo76np778i1dl5opieswiovu8bhsln 2 317452 2816041 2693071 2026-06-16T23:43:14Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816041 wikitext text/x-wiki __NOTOC__ ==9.1 - Factors Surrounding Illness== ==== Overview of Cultural Influence on Illness Behaviors ==== * '''Definition of Illness Behaviors''': These include recognizing symptoms, seeking treatment, and adhering to prescribed treatments. Cultural factors influence these stages significantly. Involve the ways that individuals monitor internal states. * '''Sociodemographic Variables''': Age, gender, ethnicity, and socioeconomic status (SES) shape how individuals perceive and respond to illness. * '''Mistrust and Disparities''': Groups like Black Americans and trans women exhibit mistrust towards the healthcare system, impacting their utilization of health services. ==== Key Cultural and Demographic Insights ==== * '''Black Americans''': Mistrust in healthcare systems contributes to racial health disparities. They may consult physicians more if it’s convenient or if illness risk feels high but rely less on social consultation. * '''Latinx Populations''': Underutilize inpatient mental health services and exhibit better healthcare usage when treated by Spanish-speaking providers. * '''Asian Americans''': Display high usage of complementary and alternative medicine (CAM), underutilize Western health services, and report significant reliance on dual health systems. ==== Traditional and Alternative Medicine ==== * '''CAM Use Among Minority Groups''': Asian Americans, particularly Chinese Americans, show a preference for CAM. Indian populations favor Ayurvedic medicine, emphasizing the balance of bodily humors. * '''Dual Health Systems''': Minority groups often combine traditional and Western medicine, showing the persistence of cultural health practices. ==== Acculturation’s Dual Role ==== * '''Positive and Negative Effects''': ** Increased acculturation can lead to better access to Western healthcare but may also result in poorer health outcomes due to unhealthy lifestyle adaptations. ** Acculturation influences provider-patient dynamics, with shared language and cultural identity improving patient outcomes. ==== Illness Representation and Perception ==== * '''Concept''': Illness representations refer to the cognitive framework individuals use to interpret and react to illness. * '''Activation''': When illness representations are triggered, behaviors align with perceived symptoms. * '''CSM Framework''': The Commonsense Model (CSM) focuses on how individuals actively make sense of illness, influenced by personal, cultural, and environmental factors. ==== The Commonsense Model of Illness Behavior ==== * '''Core Principles''': *# People act as problem-solvers in response to health threats. *# Illness management is shaped by sociocultural attitudes and beliefs. * '''Key Measures''': ** '''Illness Perception Questionnaire-Revised (IPQ-R)''': Explores symptoms, consequences, and control. ** '''Brief Illness Perception Questionnaire (Brief IPQ)''': Evaluates illness-related components in nine items. * '''Research Insights''': ** Illness representations directly affect health outcomes. ** Coping strategies mediate between illness representations and outcomes. ==== Implications for Healthcare Systems ==== * '''Provider Ethnic Identity and Acculturation''': These factors influence care quality and cultural competence in treating diverse populations. * '''Underutilization of Services''': Some minority groups face barriers like under-referrals by health practitioners, impacting health equity. * '''Mistrust in Healthcare System''' due to medical errors. ==== Conclusion and Future Directions ==== The text highlights the intricate interplay between culture, demographics, and healthcare utilization. Models like the CSM provide a structured approach to understanding illness behaviors, emphasizing the need for culturally sensitive healthcare systems. Addressing mistrust, enhancing cultural competence among providers, and integrating traditional medicine can bridge gaps in health equity. == 9.2 - Recognizing Symptoms == '''Physical Symptoms and Immediate Action:''' Physical injuries like those from accidents or sports are often addressed immediately due to visible symptoms, such as limited movement or severe pain. However, less obvious symptoms or non-life-threatening conditions are frequently ignored or delayed in receiving medical attention. '''Psychological Factors Influencing Delays:''' * '''Misattributions:''' ** People sometimes downplay serious symptoms, as humorously illustrated in ''Monty Python and the Holy Grail'' with the knight who dismisses losing limbs as “just a flesh wound.” ** Misattribution can lead to delayed care, as individuals wrongly link symptoms to less serious causes. * '''Confirmation Bias:''' ** Believing symptoms aren’t serious leads to selectively seeking information that supports this belief. ** For instance, skin discoloration may be dismissed as harmless, reinforcing a false sense of health. ** This bias can result in underestimating the need for medical care and overestimating personal resilience. * '''Attributions of Symptoms:''' ** Cultural and personal beliefs influence how symptoms are interpreted. ** For example, spiritual or cultural attributions, like considering epilepsy a sign of shamanism, can affect treatment choices. * '''Self-Fulfilling Prophecies:''' ** Misattributing symptoms, such as attributing tiredness to a cold rather than lack of sleep, can amplify anxiety and reinforce false conclusions. * '''Physician Biases:''' ** Physicians may also misattribute physical symptoms to psychological issues, leading to underdiagnosis and mistreatment. *'''Illusory Correlation:''' **People believe they are right about an association between variables more often than they actually have. '''Personality Traits and Symptom Recognition:''' * High anxiety and neuroticism often result in heightened sensitivity to symptoms and frequent health complaints. * Conversely, optimism and high self-esteem may delay care as individuals believe in their body’s ability to heal. * Those with health anxiety (hypochondriasis) may excessively seek medical attention for minor changes. '''Cultural and Social Influences:''' * Norms around health and medical decision-making vary by culture, affecting symptom reporting and treatment-seeking behaviors. * Gender differences: Women often report more symptoms due to higher '''private body consciousness,''' high alert to internal body states. '''Individual Differences in Health Involvement:''' * Patients vary in their desire for information and involvement in care, which influences satisfaction and outcomes. Matching physician communication style with patient preferences can improve care experiences. '''Key Illness-Specific Factors:''' * Illness type influences symptom reporting and treatment delays. For example, lung cancer symptoms may be ignored due to stigma or lack of knowledge. '''Informational Involvement''': * When people want to know more concerning their illnesses and treatment plans Understanding these psychological, cultural, and individual factors can help address delays in seeking medical treatment and improve health outcomes. == 9.3 - Seeking and Adhering to Treatment == * '''Illness Representations''': Personal beliefs and understandings of a health problem. * '''Exploratory Model (Kleinman, 1980)''': A framework describing three overlapping health-care systems: *# '''Popular Sector''': Culturally based family and personal health beliefs. *# '''Folk Sector''': Cultural traditions and specialized non-professional healers. *# '''Professional Sector''': Western medical professionals and systems. ---- * '''Treatment Decisions''': ** People primarily consult the popular sector (family or personal beliefs). ** Cross-cultural illness interviews involve questions about the cause, onset, impact, treatment preferences, and fears. * '''Cultural Influences on Treatment''': ** '''Collectivist Cultures''': Families or communities play a central role in caregiving (e.g., Middle Eastern, Native American). ** '''Stigma''': Disabilities or illnesses may be hidden to protect family reputation (e.g., tuberculosis in Haitian culture). ** '''Spiritual Healers''': Common first choice in cultures with strong spiritual beliefs (e.g., curandero, root-worker). ---- ==== Barriers to Seeking Treatment ==== * '''Misinterpretation of Symptoms''': People may underestimate or misattribute symptoms (e.g., indigestion vs. heart attack). * '''Social Concerns''': Fear of false alarms, inconvenience, or troubling others. * '''Financial Barriers''': Lack of health insurance or cost concerns. ---- ==== Delays in Treatment ==== # '''Appraisal Delay''': Time to recognize symptoms. # '''Illness Delay''': Time between recognizing symptoms and deciding to seek care. # '''Use Delay''': Time between deciding to seek care and obtaining it. ---- ==== Triggers for Seeking Treatment (Zola, 1964) ==== # Severity and visibility of symptoms. # Pain level or interference with daily life. # Interpersonal crises affecting relationships. # Social interference (e.g., work or vacation disruption). # Social sanctions (e.g., employer pressures). ---- ==== Cultural and Demographic Patterns ==== * '''Gender and Age''': Women and elderly seek treatment more often than men and younger individuals. * '''Ethnicity''': ** Non-European Americans often rely on lay-referral systems (family/friends). ** Close-knit communities may promote or hinder seeking care. * '''Socioeconomic Status (SES)''': ** SES affects treatment decisions and access, and is the largest cultural factor that predicts treatment seeking. ** Low-income groups may delay treatment due to costs or mistrust of the system. ---- ==== Cultural Examples ==== * '''Middle Eastern Cultures''': Hidden disabilities to avoid societal stigma. * '''Haitian American Culture''': Tuberculosis patients isolated outside the home. * '''Orthodox Jewish Communities''': Weak eyes may go untreated to enhance family marriage prospects. ---- === Hospital Setting === * '''Types of Hospitals''': For-profit and nonprofit hospitals often compete for patients in cities, benefiting patients with discounts/special services. * '''Challenges''': ** Health care is viewed as a business, leading to patient dissatisfaction. ** Millions remain uninsured despite the Affordable Care Act. ** Hospital visits often involve lengthy waiting periods, form-filling, and procedural delays. * '''Physician Interaction''': ** Physicians spend limited time with patients due to quotas and insurance demands. ** Poor patient–physician communication is a significant dissatisfaction factor. === Stress and Environment === * '''Stress Factors''': Long waits, noise, and intimidating equipment. * '''Improvements''': ** Better lighting, reduced noise, and nature scenes in waiting areas lower stress. ** Avoiding uncontrollable daytime TV in waiting rooms helps reduce anxiety. === Staff Relations === * '''Diversity''': Hospitals employ culturally diverse staff (e.g., Indian, Chinese, Middle Eastern doctors; Mexican, Filipino nurses). * '''Cultural Influences''': ** Male doctors from some cultures may show sexism. ** Asian/Latinx nurses may avoid conflict or questioning authority. * '''Religious Conflicts''': Staff may refuse procedures like blood transfusions or abortions due to religious beliefs. * '''Gender Dynamics''': ** Female doctors may face intimidation from male counterparts. ** Male nurses from patriarchal cultures may struggle with hierarchical roles. === Adherence to Treatment === * '''Definition''': Adherence = following prescribed treatments (e.g., timing, dosage, lifestyle changes). * '''Impact''': ** Poor adherence costs $100–$300 billion annually and worsens health outcomes. ** Specific risks: kidney transplant failures, AIDS-related fatalities. * '''Rates''': ** ~33% nonadherence for acute illnesses, ~55% for chronic illnesses. ** Influenced by treatment complexity and personal preferences. === Cultural & Practical Barriers === * '''Cultural Factors''': ** Dietary restrictions (e.g., Ramadan fasting, kosher laws) may conflict with treatment plans. ** Different cultural beliefs about rest periods and medical authority affect adherence. * '''Creative Nonadherence''': Patients modify treatments (e.g., skipping doses or overusing asthma sprays). === Innovations to Improve Adherence === * '''Technology''': Digital tools like pill monitors and automated programs improve compliance. * '''Parental Involvement''': Supervised adherence in children yields better outcomes. ==== Conclusions ==== * Health-seeking behaviors are shaped by cultural norms, financial barriers, social triggers, and personal beliefs. * Psychological and cultural barriers can delay treatment, increasing severity at the time of care. * Socioeconomic and demographic factors significantly impact health care access and choices. == 9.4 - Patient-Practitioner Interactions == === Patient’s Journey and Interaction with Health Care === * '''Initial Steps''': ** Patients must navigate insurance, hospital paperwork, and registration. ** Quality of interaction with doctors impacts recovery and adherence. === Patient–Practitioner Interaction Models (Szasz & Hollender, 1956): === # '''Active-Passive Model''': #* Doctor makes decisions; patient has little/no input. # '''Guidance-Cooperation Model''': #* Doctor leads; patient answers questions but doesn’t decide on treatment. # '''Mutual Cooperation Model''' (Optimal): #* Collaborative planning between doctor and patient. === Communication Types and Barriers === * '''Communication Styles''': ** Biomedical: Focus on jargon; closed-ended questions. ** Consumerist: Patient-driven discussion. * '''Cultural Factors''': ** '''Individualism vs. Collectivism''': *** Collectivists prioritize harmony, may withhold some info. *** Individualists prefer directness, potentially causing frustration in mismatched interactions. ** '''Language Challenges''': *** Jargon, technical terms, and word meanings vary (e.g., “positive” results). *** Misunderstandings arise from differences in English dialects and other languages. * '''Doctor’s Role in Communication Issues''': ** Interruptions (avg. after 18 seconds per Beckman & Frankel, 1984). ** Overuse of jargon or “dumbing down” messages. ** Insufficient explanation of uncertain outcomes. * '''Patient’s Role in Communication Issues''': ** Anxiety leads to incomplete symptom reporting. ** SES and language differences affect understanding. === Key Influences on Patient–Practitioner Communication === * '''Cultural Dimensions''': ** Varying use of small talk across cultures. ** Interpretation of symptoms and social norms differ. * '''Uncertainty and Decision Making''': ** Increasing trend for shared decisions. ** Patients often struggle with probabilities and risk communication. === Gender Bias and Stereotypes === * '''Gender Bias''': ** Stereotyping based on gender leads to differential treatment. * '''Cultural Sensitivity''': ** Symptoms can have private connotations across cultures, affecting reporting. === Importance of Intercultural Skills === * '''Training for Practitioners''': ** Incorporate intercultural and patient-centered communication skills. ** Emphasize empathy and adaptability to improve care quality. # '''Stereotypes''': Widely held, oversimplified beliefs that people have certain characteristics due to group membership. #* Examples: #** Asians: Good at math. #** Women: Bad at math. #** Middle Eastern/Indian Americans: Vocal about pain. #** Asian Americans: Quiet and stoic. #** Mexican Americans: Large families. # '''Cultural Competency''': The combination of culturally appropriate attitudes, knowledge, and skills to deliver effective healthcare to diverse groups. #* '''Cultural Awareness''': Appreciation of external signs of diversity (e.g., arts, dress, food). #* '''Cultural Sensitivity''': Avoiding offensive actions/statements based on cultural differences. # '''Perceived Discrimination''': Patient perception of biased or unequal treatment based on cultural or gender differences. # '''Gender Bias in Healthcare''': Unequal treatment based on gender, leading to disparities in diagnosis, procedures, and outcomes. #* Examples: #** Women less likely to receive ICU care or heart-related devices. #** Men more likely to be recommended knee surgery. ---- === Key Information === ==== Stereotyping in Healthcare ==== * Stereotypes influence communication and decisions in patient care. * Harmful effects: ** Generalizations can lead to poor care or malpractice. ** Stereotypes based on race, gender, or age often dictate treatment decisions. ==== Examples of Healthcare Disparities ==== * '''Racial Bias''': ** Black, Latinx/e, and low-SES patients receive less information and care (IOM, 2002). * '''Gender Bias''': ** Heart disease devices used more in men than women despite similar symptoms (Curtis et al., 2007). ** Older women face shorter ICU stays and higher mortality risks (Fowler et al., 2007). * '''Ethnic Stereotypes''': ** Language-based discrimination impacts help-seeking behavior (Spencer & Chen, 2004). ==== Cultural Competency ==== * Essential for effective communication and care. * Poor cultural competency leads to: ** Miscommunication. ** Lower patient trust and satisfaction. ** Discrimination. * '''Measurement Tools''': ** Healthcare Provider Cultural Competency scale (Lucas et al., 2008). ==== Impact of Discrimination ==== * Language barriers result in patients seeking informal care instead of formal treatment. * Women more likely than men to seek emotional support through informal means. ---- === Actionable Steps === * Develop cultural competency through awareness, sensitivity, and skills. * Avoid relying on stereotypes; assess individual patient needs. * Educate healthcare providers on the impact of bias and disparities. * Use standardized tools to evaluate cultural competency and improve patient satisfaction. [[Category:Atcovi/Health Psychology]] l3ubu7j18wagrxaxz94d9bb4no3m85r 2 317470 2816042 2693977 2026-06-16T23:43:25Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816042 wikitext text/x-wiki == 10.1 - The Experience of Pain and Its Cultural Variations == * '''Pain''' - Defined as "an unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage". Highlights study where armor in the virtual world reduced subjective pain, despite an actual sting in the arm. **'''Nociception''' - Activation of specialized nerve fibers and receptors in response to harmful stimuli (most basic level of pain). Signals occurence of tissue damage, but does not ''automatically'' lead to pain as subjective interpretation takes place. Accompanied by cognitive, behavioral, and affective states. **Important to take a '''biopsychosocial''' approach when dealing with pain, and is necessary for survival. '''Cultural Variations in the Experience of Pain''' * Boys are taught to hide the pain, while girls express pain. * Majority of studies show pain intensity and unpleasantness are not different between the two genders. Women experience more social support for their pain. * Pain experiences and treatments differ significantly across ethnic groups due to cultural, socioeconomic, and psychological factors. Research shows that Black Americans, Asian Americans, and Latinx/e individuals generally exhibit higher pain sensitivity and lower tolerance than non-Hispanic European Americans. Black Americans, in particular, face consistent disparities in pain treatment, regardless of pain type or setting, while Latinx/e people experience disparities in specific contexts. Cultural barriers, such as language differences and stress from socioeconomic challenges, further influence pain management and perception. Studies also highlight unique concerns in Asian populations regarding cancer pain and suggest that ethnic identity mediates some differences in pain experiences among groups. While individual variations exist within cultures, limitations in research methods, such as small or non-random samples, complicate the understanding of these phenomena. Future research should aim for larger, more diverse samples and explore traditional healing practices to enhance the understanding of cultural differences in pain. '''Typologies and Biology of Pain''' * Pain is a complex experience described through various terms and classifications, including '''acute pain''' (short-term) and '''chronic pain''' (long-term). Chronic pain can further be categorized as '''malignant''' (associated with diseases like cancer) or '''noncancer pain''' (e.g., lower back pain). Acute pain typically resolves within three months, while chronic pain may persist for years and is often considered a disease. Pain is also classified by its origin, such as psychogenic (psychological), neuropathic (pure nociception), or somatic (physiological without tissue damage). Pain involves four physiological processes: '''transduction''', where stimuli (chemical, mechanical, or thermal) are converted into nerve signals; '''transmission''', the relay of these signals to the central nervous system; '''modulation''', which controls the pain signals between brain regions; and '''perception''', where these signals result in the subjective experience of pain. Key structures in these processes include sensory receptors, afferent fibers, the spinal cord, and various brain regions such as the thalamus and cortex. This intricate system highlights the multidimensional nature of pain, emphasizing the need for precise classification and measurement methods. == 10.2 - Measuring Pain == * Pain is a multifaceted and subjective experience, making it challenging to measure objectively. Physical damage, like a broken limb, may cause varying pain levels in different individuals, and language barriers can further complicate pain assessment. The Initiative on Methods, Measurement, and Pain Assessment in Clinical Trials (IMMPACT) highlights four key areas for assessing pain: intensity, physical functioning, emotional functioning, and overall well-being. Pain is now widely recognized as a vital sign, alongside temperature, pulse, blood pressure, and respiration. Efforts like the National Pain Strategy aim to improve pain measurement through large-scale surveys and healthcare data. However, immediate measurement remains difficult in cases where language barriers exist or when assessing non-verbal individuals, such as young children, underscoring the need for innovative and inclusive assessment methods. '''Basic Pain Measures''' Hospitals employ various tools and techniques to assess pain across diverse cultural and demographic groups. Visual tools, such as pain scales with illustrative faces ranging from smiling (no pain) to frowning (extreme pain), are widely used. These scales often include multilingual instructions for accessibility. Numeric (NRS), verbal (VRS), and visual analog (VAS) scales are also common, though each has unique challenges. For instance, older adults or individuals from certain cultural backgrounds may find numerical pain ratings abstract. Advanced tools, like the '''McGill Pain Questionnaire (MPQ)''' and the '''Multidimensional Pain Inventory (MPI)''', delve into sensory, emotional, and evaluative aspects of pain. Innovative approaches such as '''Ecological Momentary Assessment (EMA)''' allow patients to report pain in real-time using digital devices, reducing reliance on memory. Behavioral observation also provides critical insights, especially for non-verbal patients, using tools like the '''UAB Pain Behavior Scale'''. Psychological factors, such as catastrophizing, are evaluated with specialized questionnaires, linking mental outlook to pain perception. While physiological measures like EEG and EMG exist, their efficacy in pain assessment remains limited, highlighting the importance of a biopsychosocial approach to understanding and managing pain. == 10.3 - Theories of Pain == '''Early Physiological and Psychological Approaches''' The history of pain theories reveals humanity's evolving attempts to understand and address this universal experience. Ancient beliefs attributed pain to supernatural causes, such as evil spirits or divine will. By 500 BCE, Greek philosophers viewed pain as a consequence of ''irrational thinking'', integrating it with their rational approach to human experiences. In 1664, '''Descartes''' introduced one of the earliest mechanistic explanations, proposing that pain resulted from specific stimuli transmitted through nerves to the brain, prompting a coordinated response. This unidimensional "specificity" theory was later formalized by '''Von Frey''' in 1894, suggesting dedicated pain receptors and pathways. Around the same time, '''Goldschneider’s "pattern theory"''' argued that pain arises from the integration of nerve impulses rather than specific receptors. Neither theory fully explained individual differences in pain perception or variability in treatment outcomes. '''Engel’s 1959''' "pain-prone personality" model introduced psychological influences, though it lacked empirical support. Later advancements included the '''gate control theory''' (Melzack & Wall, 1965), emphasizing the interaction between psychological and physiological factors, and '''Melzack's neuromatrix theory''' (1999), which proposed a brain-centered model incorporating genetics, emotions, and experience. Modern perspectives recognize pain as a complex interplay of biological, psychological, and social factors, reflecting the biopsychosocial approach to health. '''Biopsychological Theories of Pain''' '''Theories and Models of Pain''' # '''Early Theories''' #* Early models were primarily physiological or psychological. #* Cognitive-behavioral model: Suggests learned expectations condition pain experiences (e.g., fear of dental visits). #* Diathesis-stress model: Links physiological predispositions (e.g., low pain thresholds) with psychological factors. # '''Gate Control Theory of Pain (GCT)''' #* Proposed by Melzack & Wall (1965), it integrates biopsychosocial aspects. #* Pain signals from receptors (A-beta, A-delta, C fibers) pass through a "gate" in the dorsal horn of the spinal cord. #** '''A-beta fibers''': Large, myelinated, and inhibit pain by closing the gate. #** '''A-delta & C fibers''': Small, unmyelinated, and open the gate, intensifying pain. #* The balance of fiber activity influences pain intensity and duration. #* Chronic pain may occur when large fibers (A-beta) are deactivated. # '''Pain Management Insights''' #* Counterirritation (e.g., scratching or electrical stimulation) activates A-beta fibers, temporarily closing the gate. #* Descending pathways from the brain modulate pain: #** Positive emotions reduce pain by closing the gate. #** Negative emotions increase pain by keeping the gate open. # '''Psychological Factors in Pain''' #* Depression, anxiety, and personality disorders are linked to increased pain. #* Internal locus of control correlates with less severe pain perception. #* Cognitive appraisals and beliefs about pain significantly affect tolerance and severity. # '''Learning and Behavior in Pain''' #* '''Social learning''': Observing others’ reactions influences pain expectations (e.g., Bandura’s findings). #* '''Operant conditioning''': Reinforced pain behaviors (e.g., attention from others) can amplify pain perception. #* '''Classical conditioning''': Associating pain with specific stimuli (e.g., dentist visits) increases pain anticipation. # '''Innovative Pain Treatments''' #* Virtual Reality (VR): Emerging tools like VR exercises help alleviate pain by redirecting focus. #* Psychosocial strategies: Changing attitudes and beliefs about pain can significantly reduce its impact. == 10.4 - Pain Management Techniques == There are three main categories of pain management: physiological, psychological, and self-regulation techniques. These methods vary based on individual pain thresholds (the level at which pain is perceived) and tolerance (the level beyond which pain becomes unbearable). Cultural and gender differences primarily affect pain tolerance. '''Key Techniques:''' * '''Physiological:''' Medications like aspirin and morphine, and interventions like acupuncture or electrical nerve stimulation. * '''Psychological:''' Techniques such as relaxation, guided imagery, hypnosis, and distraction. * '''Self-Regulation:''' Long-term approaches like biofeedback, meditation, and self-management programs for chronic pain. These methods aim to achieve analgesia (pain relief) and are applied differently depending on whether the pain is acute or chronic. '''Physiological Treatments''' # '''Chemical Methods:''' #* '''Over-the-Counter Medications:''' Aspirin, acetaminophen, and ibuprofen are common for minor pains. #* '''Prescription Drugs:''' Includes opioids (e.g., morphine, oxycodone) for severe or chronic pain. Opioids act on brain receptors but risk addiction and tolerance. #* '''Cultural and Usage Trends:''' Disparities exist in opioid prescriptions across ethnic groups. # '''Acupuncture:''' #* Based on stimulating specific points to restore energy flow or close pain gates. #* Shown to release endorphins and provide analgesia, though effectiveness varies and may involve placebo effects. # '''Surgery:''' #* Nerve severing or brain lesioning can temporarily reduce pain but often fails long-term due to nerve regrowth. # '''Other Physiological Methods:''' #* '''Non-Medication Treatments:''' Use of heat, cold, vibrations, and decompression systems for localized pain relief. #* '''Stress-Induced Analgesia (SIA):''' Exercise can trigger natural pain relief through endogenous opioid release, like the “runner’s high.” These methods cater to both acute and chronic pain management but vary in effectiveness and risk profiles. '''Psychological Treatments''' # '''Expectations and Placebo Effect:''' #* Beliefs about pain or treatment can influence pain perception. Positive expectations often result in pain relief. # '''Psychological States and Cognitive Styles:''' #* Negative emotions like anxiety and depression intensify pain. #* Cognitive biases (e.g., catastrophizing or learned helplessness) can worsen pain but are modifiable through therapy. # '''Distraction:''' #* Activities like reading, watching TV, or listening to music can redirect focus from pain. #* Techniques like guided imagery and cognitive distraction are effective, especially in acute pain scenarios. # '''Hypnosis:''' #* Combines relaxation and suggestion to reduce pain perception. Self-hypnosis shows promise for chronic pain relief. # '''Cognitive and Relaxation Methods:''' #* Biofeedback, meditation, mindfulness, and relaxation reduce muscle tension and anxiety, aiding pain management. #* Massages and guided imagery are also effective. # '''Environment and Visual Influence:''' #* Natural views or simulations (e.g., greenery, art, or nature sounds) can lower pain and enhance recovery. # '''Virtual Reality (VR):''' #* Immersive VR environments distract patients from pain, mimicking the analgesic effects of nature or interactive distractions like video games. These techniques show how psychology and environment significantly influence pain perception and management. '''Self-Management of Chronic Pain''' * '''Self-Management Programs:''' ** Shift responsibility for change to the patient rather than solely relying on doctors or medical staff. ** Emphasize psychological and behavioral changes over medication or physical procedures, reducing side effects. * '''Focus Areas:''' ** Address emotional, cognitive, and sensory experiences of pain. ** Consider pain-related behaviors and social consequences, such as daily activities and relationships. ** Modify cognitive processes like attention focus, pain-related memories, coping strategies, expectations, and self-perceptions. * '''Goals of Self-Management Programs:''' ** Teach skills to redirect attention away from pain. ** Enhance physical fitness and increase daily physical activity. ** Provide coping mechanisms for severe pain episodes without relying on medication. ** Equip patients to manage emotions like depression, anger, and anxiety. ** Reduce stress, interpersonal conflicts, and unhealthy behaviors. * '''Program Structure:''' ** Begins with an intensive interview and evaluation of medical history, pain, and functional status. ** Patients collaborate with staff to set program goals and sign a contract to work toward them. ** Includes education, skills training, relaxation techniques, and cognitive-behavioral strategies to change maladaptive thoughts and actions. ** Covers relapse prevention and follow-up to ensure long-term progress. * '''Broader Perspective on Pain Management:''' ** Pain is a universal experience, and coping methods vary across cultures. ** A comprehensive understanding of diverse pain management techniques enhances adaptability and resilience. [[Category:Atcovi/Health Psychology]] k5e2pyn0vyljvoqdcfi96jkoven2js6 2816048 2816042 2026-06-16T23:46:59Z Atcovi 276019 2816048 wikitext text/x-wiki == 10.1 - The Experience of Pain and Its Cultural Variations == * '''Pain''' - Defined as "an unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage". Highlights study where armor in the virtual world reduced subjective pain, despite an actual sting in the arm. **'''Nociception''' - Activation of specialized nerve fibers and receptors in response to harmful stimuli (most basic level of pain). Signals occurence of tissue damage, but does not ''automatically'' lead to pain as subjective interpretation takes place. Accompanied by cognitive, behavioral, and affective states. **Important to take a '''biopsychosocial''' approach when dealing with pain, and is necessary for survival. '''Cultural Variations in the Experience of Pain''' * Boys are taught to hide the pain, while girls express pain. * Majority of studies show pain intensity and unpleasantness are not different between the two genders. Women experience more social support for their pain. * Pain experiences and treatments differ significantly across ethnic groups due to cultural, socioeconomic, and psychological factors. Research shows that Black Americans, Asian Americans, and Latinx/e individuals generally exhibit higher pain sensitivity and lower tolerance than non-Hispanic European Americans. Black Americans, in particular, face consistent disparities in pain treatment, regardless of pain type or setting, while Latinx/e people experience disparities in specific contexts. Cultural barriers, such as language differences and stress from socioeconomic challenges, further influence pain management and perception. Studies also highlight unique concerns in Asian populations regarding cancer pain and suggest that ethnic identity mediates some differences in pain experiences among groups. While individual variations exist within cultures, limitations in research methods, such as small or non-random samples, complicate the understanding of these phenomena. Future research should aim for larger, more diverse samples and explore traditional healing practices to enhance the understanding of cultural differences in pain. '''Typologies and Biology of Pain''' * Pain is a complex experience described through various terms and classifications, including '''acute pain''' (short-term) and '''chronic pain''' (long-term). Chronic pain can further be categorized as '''malignant''' (associated with diseases like cancer) or '''noncancer pain''' (e.g., lower back pain). Acute pain typically resolves within three months, while chronic pain may persist for years and is often considered a disease. Pain is also classified by its origin, such as psychogenic (psychological), neuropathic (pure nociception), or somatic (physiological without tissue damage). Pain involves four physiological processes: '''transduction''', where stimuli (chemical, mechanical, or thermal) are converted into nerve signals; '''transmission''', the relay of these signals to the central nervous system; '''modulation''', which controls the pain signals between brain regions; and '''perception''', where these signals result in the subjective experience of pain. Key structures in these processes include sensory receptors, afferent fibers, the spinal cord, and various brain regions such as the thalamus and cortex. This intricate system highlights the multidimensional nature of pain, emphasizing the need for precise classification and measurement methods. [[Category:Atcovi/Health Psychology]] kbf4dqn68m69h31jtu2vnxutqhcf5x0 2 317631 2816035 2693891 2026-06-16T23:41:50Z Atcovi 276019 [[Category:Atcovi/Health Psychology]] 2816035 wikitext text/x-wiki Questions to answer about your infographic: 1. Why do you think this infographic will work to change the health behavior of your target population? Think about the theories of behavior change. 2. Explain how the infographic and resulting behavior change will benefit the larger community. Helpful Hints: • Do not just list the symptoms, causes, and treatment of the health problem. This is boring. No one will read it, and certainly no one is going to change their behavior by such a bland infographic. • Consider what will appeal to your target population. Think about what will make your target population change their behavior. • Make sure that all of the steps in the project are related to one another. You need to create an infographic that will change the behavior (Step 3) of your target population (Step 2) in order to reduce the likelihood that they will have the health problem you identified in Step == Notes from https://www.youtube.com/watch?v=uQXf_d5Mgjg == === What makes an infographic effective? === * Engaging * Illustrative * No walls of text and data Steps: # Goal for Infographic (#1) - Solves a burning problem: how can we migitate diabetes? # Collect data for infographic (#2) - See step #2 of infographic # Visualize data # Lay out your infographic using Canvas # Add style == Questions == * Pacific Islanders like red, as it symbolizes prosperity (https://www.mockett.com/blog/blog-2024-asian-heritage.html) '''Question''': Why do you think this infographic will work to change the health behavior of your target population? Think about the theories of behavior change. '''Answer:''' '''Question''': Explain how the infographic and resulting behavior change will benefit the larger community. '''Answer:''' [[Category:Atcovi/Health Psychology]] n9q9fzfccxuzfa6n78zo6wduu0v11kr 2 320737 2816093 2815787 2026-06-17T09:20:59Z Tommy Kronkvist 31941 User statistics. 2816093 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|310px|The intracanopy of a Wild Service Tree, i.e. <small>''Torminalis glaberrima'' (Gand.) Sennikov & Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).</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 (June 17, 2026), I've made just over 393,700 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}} k489mcmvh0xc0ewpj7n9ttql1hha4tr 0 321701 2815992 2811306 2026-06-16T18:17:17Z Nintendofan885 2887676 fix typo 2815992 wikitext text/x-wiki :''This discusses an interview 2025-05-08 with Seth Stern<ref name=Stern><!--Seth Stern-->{{cite Q|Q134333839}}</ref> and Lauren Harper<ref name=Harper><!--Lauren Harper-->{{cite Q|Q134371468}}</ref> about the Freedom of the Press Foundation. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2025-05-17 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:Freedom of the Press Foundation says.webm|thumb|2025-05-08 interview with Seth Stern<ref name=Stern/> and Lauren Harper<ref name=Harper/>, Director of Advocacy and Daniel Ellsberg Chair on Government Secrecy, respectively, for Freedom of the Press Foundation]] [[File:Freedom of the Press Foundation says.ogg|thumb|29:00 mm:ss of excerpts from a 2025-05-08 interview with Seth Stern and Lauren Harper of Freedom of the Press Foundation.]] Seth Stern,<ref name=Stern/> Director of Advocacy for Freedom of the Press Foundation, and Lauren Harper,<ref name=Harper/> their Daniel Ellsberg Chair on Government Secrecy, discuss their work with Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> Freedom of the Press Foundation works to protect journalists and their sources in several ways:<ref><!--Freedom of the Press Foundation -->{{cite Q|Q5500827}}</ref> * [[w:SecureDrop|SecureDrop]]: They develop and maintain their open source whistleblower submission system to facilitate anonymous and secure communications between sources and journalists.<ref><!--Technology: Our open source software tools protect newsrooms, journalists, and their sources-->{{cite Q|Q134334311}}</ref> The project was begun in part by [[w:Aaron Swartz|Aaron Swartz]], who tragically killed himself under intense pressure from the FBI on questionable grounds. * Digital security education for news organizations.<ref><!--Digital Security Education: Explore resources, training, and other services you can use to protect your work and your sources in the digital age.-->{{cite Q|Q134335013}}</ref> * [[w:Freedom of the press in the United States#U.S. Press Freedom Tracker|U.S. Press Freedom tracker]], documenting attacks on journalists including assaults and arrests for activities that seemingly should be protected by the First Amendment.<ref><!--U.S. Press Freedom Tracker-->{{cite Q|Q134335566}}, accessed 2025-05-01.</ref> They documented 2,530 such attacks on secrecy, surveillance, and the rights of journalists and whistleblowers in the 8 years between 2017 and 2024. A third of those attacks were in the single year 2020, the last year of President Trump's first term.<ref>Click "all time" at <!--https://pressfreedomtracker.us/-->{{cite Q|Q134336764}}</ref> * Staying current on these issues.<ref><!-- The Latest: Mobilizing allies and the public to create tangible change for press freedom.-->{{cite Q|Q134337247}}</ref> One "featured issue" in the last category says, "Reform Government Secrecy", claiming that, "The U.S. classifies far too many secrets, obstructing democracy."<ref><!--Reform Government Secrecy: The U.S. classifies far too many secrets, obstructing democracy-->{{cite Q|Q134337726}}</ref> This includes "‘The Classified Catalog’ launches to track secrecy news", numerous things the Trump administration has done since 2025-01-20 to erode "the information environment in ways this country has never seen.<ref><!--‘The Classified Catalog’ launches to track secrecy news-->{{cite Q|Q134387817}}</ref> These steps changes include the following: * [[Trump ordered changes in public data|Deleted thousands of datasets from agency websites]]. * Closed agencies’ [[w:Freedom of Information Act (United States)|Freedom of Information Act]] offices.<ref>Harper (2025-04-13).</ref> * Used disappearing messaging apps and failed to preserve government records.<ref>Harper (2025-03-30).</ref> * Ordered federal health agencies to stop communicating with the public.<ref>Klippenstein (2025).</ref> * Gutted a key surveillance oversight board.<ref>Weissmann (2025).</ref> * Mass-fired inspectors general.<ref>Harper (2025-01-28).</ref> * Replaced independent, professional leadership at the National Archives with unqualified appointees.<ref>Harper (2025-02-20).</ref> * Attempted to illegally destroy agency records.<ref>Harper (2025-03-16).</ref> == US government secrecy == === Complicity in nuclear proliferation === Graves asked about claims by [[w:Richard Barlow (intelligence analyst)|Richard Barlow]] that the US State Department had clandestinely supported illegal exports of "dual use technologies" to Pakistan, without which Pakistan would likely not have nuclear weapons today -- and North Korea got some of their nuclear technology from Pakistan. Harper ageed, noting that she had previously worked with the [[w:National Security Archive|National Security Archive]]. William Burr<ref> https://nsarchive.gwu.edu/about/staff/dr-william-burr </ref> directs their "Nuclear Vault", which contains resources from their "Nuclear Documentation Project".<ref> https://nsarchive.gwu.edu/project/nuclear-vault </ref> He has led a decades-long struggle to get records that are 50, 60 and 70 years old declassified. Many documents they get are so heavily redacted that you cannot make sense out of any of it. This "makes it more difficult for current policymakers to craft effective and rational nuclear policy." === Moynihan Commission on Government Secrecy === Graves then asked about the [[w:Moynihan Commission on Government Secrecy|Moynihan Commission on Government Secrecy]] of the 1990s. Harper said "they basically said that government secrecy effectively works as a form of government regulation, because the public cannot engage in their right to selfgovern when they don't have access to this information that the government routinely overclassifies. One of the things they suggested over 30 years ago and we still haven't done is that the Senate should get involved, and the Congress should get involved in legislating on classification." Harper mentioned a couple of bills to reform government secrecy that had been introduced in the last Congress but died in committee. She hopes they will get reintroduced. === Claims of national security === Graves asked about "[[w:State secrets privilege|state secrets privilege]]". Stern said that it severely limits the ability of anyone to question a claim of national security. "We are seeing the Trump administration frequently abuse the national security flag ... whenever it wants to make exceptions to the law." Occasionally a media outlet will write a story or an editorial expressing concern about such claims, but there's no follow-up reporting. Stern noted that [[w:Donald Trump–TikTok controversy|TikTok, an app hosting millions of posters, was banned]] based on admittedly hypothetical threats that China might use it to spy on Americans. Nothing was ever proven, "and now suddenly Donald Trump has decided he doesn't want to ban TikTok anymore, because he sees a route to make money off of it, and no one's concerned about national security." Stern mentioned the [[w:2022–2023 Pentagon document leaks|discord leaks a couple years ago]]. It was a big story with plenty of reporting for two days that quoted "administration officials saying, 'The sky is falling. This is a major threat to national security. ... [P]eople are in danger for their lives.' ... Six months later, the sky is still there, and nothing has happened that these people predicted. There's no follow-up reporting." == The role of the media in sustaining the system of political corruption == Graves asked about the role of the media in sustaining the system of political corruption that threaten us and international security. Stern said, {{quote|One thing that the last few weeks have put to rest is the myth that billionaires and major conglomerates with interests far beyond the media or their news holdings can possibly run news outlets without impacting the direction of coverage -- without either directly or indirectly causing reporters to shy away from stories that might upset either business or political connections. ... [W]hen you had a president who was willing to threaten their business interests, ... they caved immediately. They settled defensible cases. The case ABC settled was very defensible. ... When I was practicing law, I defended an almost identical case involving a college professor who had been accused of sexual assault. A newspaper reported he had been accused of rape. A judge threw that out, saying essentially the terms are interchangeable. How were you damaged? Find me someone who was willing to do business with an accused sexual assailant but drew the line at an accused rapist. That person doesn't exist. There are no damages. This is a frivolous case. ... CBS is currently mediating over editing of a video interview that is even more baseless. You will not find a First Amendement law expert in the world who is not wearing a Donald Trump lapel pin who is going to tell you that that case has any legal ground whatsoever. And it's pretty much an open secret, not even a secret, that the only reason CBS is even thinking about settling this case is because it wants the Trump administration to approve its merger. Essentially they are using the legal system to launder what would otherwise be called bribes, but which are okay as long as a judge signs off on it. ... Despite all the brilliant journalists who work for corporate media outlets -- and I'm not looking to knock anyone ... ''[[w:The New York Times|The New York Times]]'', ''[[w:The Washington Post|Washington Post]]'', ABC, CBS, they all have incredible journalists working for them. But the end product is not in the hands of those individuals. How much any particular story gets headline news treatment versus gets burried. That's not in the hands of those individuals. ... I think it's really time for people who value well reported, independent, aggressive, adversarial journalism to support independent news outlets, nonprofit news outlets. Nonprofits aren't a perfect solution. You're still subject to the whims of donors.}} Harper added that the money from the ABC settlement was reportedly "going to Trump's presidential library. But this isn't technically true. It was going to a private presidential foundation and museum, ... and those are private, effectively corporate entities with basically no campaign contribution limits ... . It's an excellent way and an excellent plact to put dark money. ... [I]t's being reported as going to something that's going to somehow enrich the public understanding of the Trump Presidency, which, of course, it won't." == Local news == Stern encourages people to "subscribe to their local papers. We've got [[w:news deserts|news deserts]] all over this country." Graves added that "local" should mean locally owned, not part of a major national chain. Stern agreed, saying that's what he meant by "local". Graves noted that was "not obvious. [[Vulture capitalists destroying newspapers|''The Denver Post'' is not a local paper anymore.]]" Seth replied, "You're exactly right." == Previous interview with Freedom of the Press Foundation == Graves previously interviewed Kirsten McCudden, Vice President of Editorial of Freedom of the Press Foundation <ref><!--Kirstin McCudden-->{{cite Q|Q134341766}}</ref> not quite two years ago on 2023-07-18.<ref><!-- Freedom of the Press Foundation works to improve news and democracy-->{{cite Q|Q134341296}}--></ref> == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. We describe here briefly the motivation for this series. [[Great American Paradox|One major contributor to the dominant position of the US in the international political economy]] today may have been the [[w:Postal Service Act|US Postal Service Act of 1792]]. Under that act, newspapers were delivered up to 100 miles for a penny when first class postage was between 6 and 25 cents. [[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.”<ref>Tocqueville (1835, p. 93).</ref> McChesney and Nichols estimated that these newspaper subsidies were roughly 0.21 percent of national income (Gross Domestic Project, GDP) in 1841.<ref>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> At that time, the US probably led the world by far in the number of independent newspaper publishers per capita or per million population. This encouraged literacy and limited political corruption, both of which contributed to making the US a leader in the rate of growth in average annual income (Gross Domestic Product, GDP, per capita). Corruption was also limited by the inability of a small number of publishers to dominate political discourse. That began to change in the 1850s and 1860s with the introduction of high speed rotary presses, which increased the capital required to start a newspaper.<ref>John and Silberstein-Loeb (2015, p. 80).</ref> In 1887 [[w:William Randolph Hearst|William Randolph Hearst]] took over management of his father’s ''[[w:San Francisco Examiner|San Francisco Examiner]]''. His success there gave him an appetite for building a newspaper chain. His 1895 purchase of the ''[[w:New York Morning Journal|New York Morning Journal]]'' gave him a second newspaper. By the mid-1920s, he owned 28 newspapers. Consolidation of ownership of the media became easier with the introduction of broadcasting and even easier with the Internet.<ref>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]]“.</ref> [[:Category:Media reform to improve democracy|This consolidation seems to be increasing political polarization and violence worldwide]], threatening democracy itself. === The threat from loss of newspapers === A previous ''Media & Democracy'' interview with Arizona State University accounting professor Roger White on "[[Local newspapers limit malfeasance]]" describes problems that increase as the quality and quantity of news declines and ownership and control of the media become more highly concentrated: Major media too often deflect the public's attention from political corruption enabled by poor media. This too often contributes to other problems like [[w:Scapegoating|scapegoating]] [[w:Immigration|immigrants]] and attacking [[w:Diversity, equity, and inclusion|Diversity, equity, and inclusion]] (DEI) while also facilitating increases in pollution, the cost of borrowing, political polarization and violence, and decreases in workplace safety. More on this is included in other interviews in this ''Media & Democracy'' series available on Wikiversity under [[:Category:Media reform to improve democracy]]. An important quantitative analysis of the problems associated with deficiencies in news is Neff and Pickard (2024). They analyzed data on media funding and democracy in 33 countries. The US has been rated as a "flawed democracy" according to the [[w:Economist Democracy Index|Economist Democracy Index]] and spends substantially less per capita on media compared to the world's leading democracies in Scandinavia and Commonweath countries. They note that commercial media focus primarily on people with money, while publicly-funded media try harder to serve everyone. Public funding is more strongly correlated with democracy than private funding. This recommends increasing public funding for media as a means of strengthening democracy. See also "[[Information is a public good: Designing experiments to improve government]]". ==Discussion == :''[Interested readers are invited 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 == * <!--Lauren Harper (2025-01-28) "With inspectors general under threat, Espionage Act charges may soar"-->{{cite Q|Q134388337}} * <!--Lauren Harper (2025-02-20) "Hostile takeover at National Archives erodes our right to know-->{{cite Q|Q134388449}} * <!--Lauren Harper (2025-03-16) " It’s Marco Rubio’s party, and he’ll burn documents if he wants to-->{{cite Q|Q134388555}} * <!--Harper (2025-03-30) "The Signalgate problem nobody is talking about"-->{{cite Q|Q134387986}} * <!--Harper (2025-04-13) "Here’s how the firing of FOIA officials could hurt the DOGE audit"-->{{cite Q|Q134387841}} * <!--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.}} * <!--Ken Kippenstein (2025-01-30) " Trump administration just ordered a blackout on public communications by agencies across government, multiple officials tell me", post to X-->{{cite Q|Q134388106}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- 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)}} * <!--Andrew Weissmann (2025-01-22) " What Just Happened: What Trump’s Hobbling Of The Privacy Oversight Board Portends For Exercise Of Surveillance Powers", Just Security-->{{cite Q|Q134389408}} [[Category:Media]] [[Category:News]] [[Category:Politics]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> arpy8gbkasppbevulqtzgkofgzxlouy 0 321771 2816032 2807236 2026-06-16T23:40:20Z Atcovi 276019 /* Future Outlook */ update page name 2816032 wikitext text/x-wiki {{Article info | journal = WikiJournal of Medicine <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Azeez | orcid1 = 0009-0007-9202-4614 | first1 = Aaqib | last2 = | first2 = | last3 = | first3 = | last4 = | first4 = <!-- up to 9 authors can be added in this above format --> | et_al = <!-- if there are >9 authors, hyperlink to the list here --> | affiliation1 = Old Dominion University | correspondence1 = yonikmalik@gmail.com | affiliations = institutes / affiliations | correspondence = email@address.com | keywords = <!-- up to 6 keywords --> | license = <!-- default is CC-BY --> | abstract = This is a narrative review. }} TBD == Introduction == Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, war crimes, and economic disruptions. Sri Lanka has only recently exited the climaxes of a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis in from 2019 to 2024]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and continues to face the ripple effects of the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is "unknown", but the statistics remain alarming despite a declining trend. Suicide rates in the country during the mid-1990s were the second-highest in the world with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country to this day<ref>{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}</ref>. == Methods == [source selection process] ==Historical Development of Mental Health Services== In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model<ref name=":0">Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.</ref><ref name=":15">{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}</ref>. [pull more info from https://www.researchgate.net/publication/342354982_Development_of_civil_commitment_statutes_laws_of_involuntary_detention_and_treatment_in_Sri_Lanka_a_historical_review maybe?] === Adoption of a Western-based mental healthcare model and issuances of ordinances === In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill in the country<ref name=":0" /><ref name=":2">{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&view=article&id=6&Itemid=125&lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}</ref>. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance "[did] not seem to have contemplated treatment as a contingency to be considered"<ref name=":1">{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}</ref>. In 1840, the 1839 Ordinance was repealed and replaced by the 1840 Ordinance. The 1839 Ordinance was almost identical to the 1840 Ordinance, except the removal of two previous requirements: the requirement for official medical diagnoses of the mentally insane and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums<ref name=":3">{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}</ref>. In 1873, a third Ordinance was released. It included linguistic changes, where the term, "insane", was replaced with "of unsound mind". The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite this Ordinance being the most comprehensive outlook on mental healthcare in the country at the time, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances<ref name=":3" />. === Development of mental asylums === At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district<ref name=":0" /><ref name=":3" />. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to the institute in Borella. Overcrowding soon became an issue and patients institutionalized at the Borella mental asylum were sent to prisons across the country. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London. ]] As medical institutions were being made to house the mentally insane, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding in just one year<ref name=":0" />. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes of the mental disorders. In 1926, the Angoda Mental Hospital was established, scantily alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. === Edward Mapother's 1937 inspection of British Ceylon === Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old<ref>{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}</ref>. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon<ref name=":1" />. In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of "a prison that is neglected and dilapidated"<ref name=":1" />. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%<ref name=":1" />. Mapother offered a vivid and grim account of the hospital in his reports: <blockquote> The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie<ref name=":0" />.</blockquote>Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff<ref name=":0" /><ref name=":1" />. On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals, which was the suggestion of a "Visiting Committee". This committee was tasked to "meet at the hospital, carry out inspections, and make recommendations" to the Executive Committee of Health<ref name=":1" />. The Government realized that deficiencies in their mental healthcare system could prove to be "costly" for their reputation. Mapother was reportedly enraged when he found out. Mapother intended to contact the Secretary of State regarding the "distortion" of his plans, but was interrupted by events preluding to [[w:World_War_II|World War II]]<ref name=":1" />. Mapother passed away on March 20, 1940, without materializing his follow-up plans. === Post-Mapother developments and further innovations === [[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. |right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of massive renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as protectional therapy expanded to [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]]<ref name=":4">{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}</ref>. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital<ref name=":0" />. In 1948, British Ceylon was granted independence from the British after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were still ongoing. The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country<ref name=":5">{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}</ref>. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years<ref name=":4" />. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967<ref name=":5" />. After 1960, mental health services were being established beyond the capital to other cities in the country<ref name=":2" />. In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] began a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a "1:500,000 to 1000,000" ratio of psychiatrists to patients on "most occasions"<ref name=":0" />. === Mental Disease Ordinance of 1956 === In 1956, the 1873 Ordinance was revised a second time and renamed the "Mental Disease Ordinance of 1956"<ref name=":5" /><ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref>. Another linguistic development is seen with the new revision as "lunacy" was replaced with "mental disease"<ref name=":6" />. The Ordinance paved the way for community-based services to be delivered to patients closer to their residences rather than solely allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population<ref name=":5" />. === Developments from the 1990s === The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&view=featured&Itemid=101&lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] who is responsible for the monitoring and implementation of mental health programs across the country<ref>{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}</ref>. As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva<ref name=":2" />. On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country beyond the capital (Colombo). The policy aimed to concisely define the rigorous standards needed to be completed for each respected medical professional, including psychiatrists and clinical psychologists<ref>{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}</ref>. The policy also included a new position, the "Medical Officer of Mental Health", who oversees and assists in the implementation of community-based mental health services<ref name=":0" />. This same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military<ref name=":8" />. In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health on what actions should be executed by the Directorate of Mental Health<ref name=":7">{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&view=article&id=9&Itemid=220&lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}</ref>. In 2008, the Angoda Mental Hospital was restructured as the National Institute of Mental Health (NIMH)<ref name=":7" />. === Modern-day Sri Lanka === [[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]] As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance<ref name=":7" />. Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are tailored towards alleviating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]] and family support for diagnosed children. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers<ref name=":7" />. [expand more on SL Gov't efforts here...] Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing "decreased incomes"<ref>Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. <nowiki>http://documents.worldbank.org/curated/en/099634104012434919</nowiki></ref>. This was churned by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]]. According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role<ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref>. == Impact of Conflicts, Terrorism, Political Instability & Natural Disasters == === Sri Lankan Civil War === The '''Sri Lankan Civil War''' was a domestic conflict that took place between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a result of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization<ref>{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}</ref><ref>{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}</ref>. Through brutal massacres, assassinations, and suicide bombings, the LTTE waged decades of terror which led to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict with the phrase ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'<ref name=":9">{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}</ref>. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources and further deepening long-term trauma and mental health deterioration in the population<ref name=":5" />. In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) performed an investigation of mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a "sense of ‘learnt helplessness’"<ref name=":5" />. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to "combat, shelling, and witnessing the death of loved ones"<ref name=":9" />. [[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians originating from the Kilinochchi and Mullaitivu Districts due to military campaigns by the Sri Lankan military (January 2009). Displaced civilians had to avoid both the atrocities committed by the LTTE and the Sri Lankan government.]] Accusation of war crimes towards [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]] have been documented by various external organizations, despite the government's attempts at removing any [https://www.youtube.com/watch?v=e_p1TfTguW0 mentions] or [https://www.youtube.com/watch?v=Vtm54Y9USEg investigations] of it<ref>See also [[w:Sexual violence in the Sri Lankan civil war]].</ref>. A 2009 HRW report stated that the Sri Lankan government assumed native Tamil population residing in war zones to be "siding with the LTTE and [therefore, were] treated as combatants", leading to indiscriminate shillings and massacres of civilians<ref>{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}</ref>. Alongside the oppression by the Sri Lankan military, the Vanni population also endured the brutal theatrics of the LTTE, which recruited men, women, and even children with minimal training, effectively rendering them cannon fodder. Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were abysmal<ref>{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}</ref>. The suicide rate in these displacement camps were three times the community-level (2002), with a ratio of 103.5 per 10,000 compared to the Sri Lankan general population's rate of 37.5 per 10,000. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse. Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a "community-based programme" which included "increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling". The MSF also advocated for restrictions of poisonous substances due to the suicide attempts, and stressed that "much more [than resettlement]" would need to be done to help alleviate the psychological pain the northern population had faced<ref>{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}</ref>. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. '''Post-war''' [[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]] The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were currently placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps tend to report more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of the husband to war, and sexist violence<ref>{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}</ref>. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the observation seen in the 2009 study<ref>{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}</ref>. Muslims in Northern Sri Lanka during the conflict also faced violence and discrimination, most notably [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), "other depressive syndromes" (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were "considerably higher" than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not show the data behind this)<ref>{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}</ref>. Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&hl=en&oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&hl=en&oi=sra Varuni de Silva], offers the camaraderie of the unit as an explanation for the discrepancy<ref>{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}</ref>. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of jarring conflict<ref>{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}</ref>. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms<ref>{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health & Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}</ref>. About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. From July 2019 to October 2020, a study was conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate - severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate - severe hopelessness<ref>{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}</ref>. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs exhibited "remarkable resilience and post-traumatic growth" after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities<ref>{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}</ref>. Findings originating from a 2019 study undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting statistics. Out of 336 participants from districts that faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited "extremely severe" symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities in the North<ref>{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}</ref>. Further research should be conducted in this field. In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that "probable depression, fatigue, aggression, and family history of mental disorder" were correlative of PTSD presence. He suggested that "screening and psychosocial intervention" were recommended avenues to alleviate CMDs of former combatants<ref>{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}</ref>. === 2004 Boxing Day Tsunami === The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to about 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage<ref name=":5" />. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka]. [[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquaters, Moratuwa).]] A survey conducted on schoolchildren (ages 8-14) in Manadkadu (Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents seem to highly correlate with PTSD development<ref>{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}</ref>. Many victims in the Jaffna area suffered with "[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD" post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in the journal ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a "large[r] (than expected) rise in [the] number of people" seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that "more psychiatric disorders" would emerge due to "very little rebuilding [efforts]" and an apparent "unfairness in the aid system".<ref>{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}</ref><ref>{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}</ref> At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] reported measures taken to assist the affected were "not coordinated" due to poor "communication systems and road [conditions]", which were disrupted by the Boxing Day tsunami. Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had "little" mental health training prior to the disaster, suggesting increased "research" and adequate "provision[ing] and training of staff" in the long-term<ref>{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}</ref>. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments. In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring "above the cut-off for anxiety" and reporting more "somatic symptoms", though researchers inferred that the PTSD rate found in the study may be influenced by war or economic hardship<ref>{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}</ref>. === 2019 Easter Bombings === The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attack was also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings regarding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups<ref>{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}</ref>. In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to "low-cost methodological" responses to children and adolescents affected by the attack as a "severe shortage" of children and adolescent mental health experts were exposed<ref>{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}</ref>. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed "doubts" in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for "culturally-sensitive" programs, a diversity in therapeutic approaches (including nature-based therapy), and "prolonged investigations" to track developments in mental health resources and impacts of implemented interventions<ref>{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}</ref>. A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces<ref>{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}</ref>. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be "made refugees again" after local protests were orchestrated against their settlement. Islamophobic aroma was "unleashed online, in the law, and on the street"<ref>{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}</ref>. Albeit its relevancy to the attacks, no in-depth mental health studies were administered on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. Literature regarding the impact of the 2019 Easter Bombings on mental health are limited and further research should be done in the field. === 2019-2024 Economic Crisis === The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced massive inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation were additional results of the economic crisis. Healthcare workers faced a barrage of mental health during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuate a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients<ref>{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}</ref>. Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a "clinical post-graduate career"<ref>{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}</ref>. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income & 1/3 of participants exhibiting "clinical levels of psychological distress"<ref>{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}</ref>. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly "general needs", while more than 50% looked into international opportunism for employment. Notably, the study reported an overall near "twofold greater" rate of depression, anxiety, and stress compared to previously conducted studies on nurses<ref>{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}</ref>. The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation by the Sri Lankan government towards a critical sector for the well-being of the country. Comprehensive mental health interventions need to be prepared and ready to implement at times of national emergencies. == Present-Day Challenges == === Ethnic tension === Despite the end of the Sri Lankan civil war and the introduction of pluralist policies, such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration, tensions amongst members of the ethnic groups still persist in the country. Evidence of these tensions was found through a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticisms, through semi-structured interviews, for "conflict transformation". A Tamil citizen of the Ratnapura community recounted that they were forced to "hide in jungles" and consume "dirty water in drainage[s]" due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about "whether [he was] Sinhalese or not". Both accounts convey interethnic tensions stemming from decade-long conflicts<ref>Jayathilaka, Aruna & Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66</ref>. Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government and their accountability of perpetrators of war crimes and information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years contribute to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. [[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]] Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that "[engages] or [tolerates] severe violations of religious freedom" in their 2024 report. The government has been criticized by human rights organizations for "disproportionately targeting religious minorities"<ref>{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}</ref>. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues. Effort to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel. Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence developed in the wake of the 2019 Easter bombings, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values should be stressed across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against threatening extremist groups promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. === Poverty === It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination<ref>{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}</ref>. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions<ref name=":8" />. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to "remain above 25 percent". Suicide has been empirically linked to economic hardships in previous studies<ref>{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}</ref>. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas<ref>{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}</ref>. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns. === Stigmas === Stigma consists of the "combined effect of prejudice, ignorance and discrimination."<ref name=":10">{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}</ref>. A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid "shame" and possible hinderances in marriage proposals. Views that mentally ill individuals were "violent" served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would have stones and "derogatory names" launched at them. A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was "improving" in their depiction of mental illness. Beliefs that illnesses are caused by "spirits" can be problematic for individuals dealing with mental health issues and serves as evidence to poor mental health awareness in the country. Mental health workers themselves believed that they were being stigmatized, as mental health is reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the contextualization of the results<ref name=":10" />. Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine. === Suicide Trends & Risk Factors === Suicide is defined as "the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome"<ref name=":11">{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}</ref>. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s due to its banning of extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022<ref>{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}</ref>. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years<ref name=":12">{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}</ref>. From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite the relevant insights into certain aspects of an average Sri Lankan's life that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates<ref name=":11" />. Further comprehensive studies on risk factors relating to suicide should be studied to assess correlations between unemployment rates and austerity measures. The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate<ref>{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}</ref>. Media potrayals of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows<ref name=":12" />. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women. == Role of Religion == According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country<ref>{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}</ref>. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life<ref name=":15" />. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] affirms hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry<ref name=":4" /><ref>{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}</ref>. Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health<ref>{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}</ref>. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have "better mental health and adapt[ed] more quickly to health problems" versus patients who weren't religious<ref>{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}</ref>. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and "make positive use of religion in [their] practice[s]"<ref>{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}</ref>. === Buddhism === 27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the "center" of overall wellness, the "precondition for a successful life"<ref name=":14">{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}</ref>. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality "cannot be directly intervened" and can only be seen through "[interactions] with society"<ref name=":14" />. Despite the ''athmaya'' (soul) being "unreachable", it can be "intervened", or treated, through the actions of the mind and body with society<ref name=":14" />. One being "psychologically ill" can affect one's spiritual being, as the participants reported in their interviews, and can be affected through "lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives"<ref name=":14" />. The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists. In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma<ref>{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}</ref>. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted. Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences<ref>{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}</ref>. === Catholicism === Catholic counseling refers to "a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church"<ref name=":13">Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.</ref> and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of "personhood"<ref name=":13" />. The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on "[[w:Evangelism|evangelization]], education, and sacramental formation". Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where "hope" and "humility" are used as the frameworks for creating spiritual resilience<ref name=":13" />. The general Christian call of "agape love and acceptance" is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness. Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the "spiritual roots of distress and moral confusion". Catholic counseling fills in this gap by integrating "psychological insights with a transcendent orientation, supporting lasting transformation and integrity"<ref name=":13" />. As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka. === Islam === The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research has revealed a 2012 case study where a 21-year-old Muslim woman was experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals<ref>{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}</ref>. Future recommendations would be to employ resources to research the foundations of Islamic psychiatry in the country, and to observe the rituals employed and their effects on patients. Studies have found that Islamic prayer can be an effective means of "support and coping"<ref name=":15" />. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with "religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”<ref>{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}</ref> Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress<ref>{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}</ref>. === Hinduism === Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework<ref name=":0" /><ref>{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}</ref>. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards<ref>{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}</ref>. === Comparative synthesis === Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These methods are practiced through karma, tawakkul, hope, and humility. Additionally, these practices are implemented in traditionally-oriented mental health care, which has been reported to be preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serving as psychologically analogous mechanisms during times of distress. In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures. == Future Outlook == '''→''' '''[[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]]''' Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A mental health legislation, drafted in 2007, includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be accepted and implemented. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated with a lack of an up-to-date legal framework, impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Based on these limitations... === Criticism of the Mental Disease Ordinance of 1956 === <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) ==Additional information== ===Acknowledgements=== Any people, organisations, or funding sources that you would like to thank. ===Competing interests=== No competing interests. ===Ethics statement=== An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section. ==References== {{reflist|35em}} [[Category:Mental health]] [[Category:Sri Lanka]] s1v39nojjbqr4zpzo15k7d7ow2w9lni 0 322847 2816020 2810713 2026-06-16T21:55:51Z Jtneill 10242 added [[Category:Motivation and emotion/Book/Therapeutic alliance]] using [[Help:Gadget-HotCat|HotCat]] 2816020 wikitext text/x-wiki {{title|Coercion and therapeutic alliance:<br>How do coercive practices in mental health care undermine trust and therapeutic relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} ;Scenario [[File:Healthcare professional writing.jpg|thumb|225px|'''Figure 1'''.Patients may feel pressured into complying by healthcare providers.]] Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a "difficult patient". Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings (see Figure 1). * What coercive practices has Sarah experienced? * What factors put Sarah at a higher risk of experiencing coercive practices? * What barriers to good therapeutic relationships are experienced by Sarah? * What could be done differently do improve her treatment outcomes? {{RoundBoxBottom}} {{expand}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What are therapeutic relationships and coercive practices? * How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships? * What are the impacts of coercive practices on therapeutic relationships and mental health outcomes? * What factors can make someone vulnerable to coercive practices? * What are the ethical implications of coercive practices, and how can they be overcome? {{RoundBoxBottom}} == Therapeutic relationships == {{ic|Include an introductory paragraph before branching into sub-sections}} === Defining therapeutic relationships === [[File:Health professional in scrubs enjoying coffee while using a smartphone in a casual work setting during morning hours.jpg|thumb|225px|'''Figure 2.''' Coercive practices can be subtle and hard to spot.]] Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland & Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018) (see Figure 2). Examples of therapeutic relationships in mental health settings include: * Nurses tending to a psychiatric care patient (Ernstmeyer & Christman, 2022). * A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland & Torrico, 2024). === Applying self-determination theory to therapeutic relationships === [[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci & Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan & Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012). * Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider. * Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle. *Relatedness: You feel understood and cared for by your mental healthcare provider. When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan & Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance. === Applying tripartite model of working alliance to therapeutic relationships === The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson & Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward. * Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties. * Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. * Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes. == Coercive practices in mental health contexts == Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner & Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). === Defining coercive practices === Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames & Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). '''Table 1.''' Examples of coercive practices {| class="wikitable" |+ !Restraint !Type of coercion !Example |- |Physical |Formal/explicit |Use of physical force to restrain a patient |- |Mechanical |Formal/explicit |Use of physical tools to restrain a patient |- |Chemical |Formal/explicit |Use of medication such as sedatives to control a patient's behaviour, rather than treating them |- |Environmental |Formal/explicit |Limiting access to environment through practices like seclusion |- |Psychological |Informal/implicit |Using psychological pressures such as implicit threats or consequences to get patients to comply |} <quiz display="simple"> {Coercive practices improve therapeutic relationships: |type="()"} + True - False {Coercive practices are always explicit: |type="()"} - True + False </quiz> === Impact of coercive practices on therapeutic relationships and mental health outcomes === Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes & Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025). '''Table 2.''' Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust {| class="wikitable" style="margin: auto;" |- !Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance |- |Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks. |- |Reduced engagement with mental health care. |Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management. |People may be unable to receive a diagnosis or to set mental health goals. |- |High incidence of trauma/trauma-related disorders, and loss of trust. |Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. |Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. |} === Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes === Coercive practices have negative impacts on those who experience them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016). Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships. Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland & Torrico, 2024) (Sharma & Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025). Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes & Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025). Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma & Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). === Ethical implications of coercive practices === Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner & Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames & Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner & Braun, 2023). == Overcoming coercive practices == Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018). ==Conclusion== In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. ==See also== * [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018) * [[wikipedia:Coercion|Coercion]] (Wikipedia) * [[wikipedia:Motivation|Motivation]] (Wikipedia) * [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia) * [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia) * [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia) * [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia) * [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) * [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia) * [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia) ==References== {{Hanging indent|1= Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., & Alegría, M. (2015). 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M., & Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753 Sashidharan, S. P., Mezzina, R., & Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350 Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version Sharma, N., & Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/ Stanhope, V., Marcus, S., & Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183 Wostry, F., Hahn, S., & Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012 }} ==External links== * [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com) * [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Control]] [[Category:Motivation and emotion/Book/Psychotherapy]] [[Category:Motivation and emotion/Book/Relationships]] [[Category:Motivation and emotion/Book/Therapeutic alliance]] oeox3rom8lo7qjzakpfpoci0m1za5pl 2816021 2816020 2026-06-16T21:56:13Z Jtneill 10242 removed [[Category:Motivation and emotion/Book/Relationships]] using [[Help:Gadget-HotCat|HotCat]] 2816021 wikitext text/x-wiki {{title|Coercion and therapeutic alliance:<br>How do coercive practices in mental health care undermine trust and therapeutic relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} ;Scenario [[File:Healthcare professional writing.jpg|thumb|225px|'''Figure 1'''.Patients may feel pressured into complying by healthcare providers.]] Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a "difficult patient". Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings (see Figure 1). * What coercive practices has Sarah experienced? * What factors put Sarah at a higher risk of experiencing coercive practices? * What barriers to good therapeutic relationships are experienced by Sarah? * What could be done differently do improve her treatment outcomes? {{RoundBoxBottom}} {{expand}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What are therapeutic relationships and coercive practices? * How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships? * What are the impacts of coercive practices on therapeutic relationships and mental health outcomes? * What factors can make someone vulnerable to coercive practices? * What are the ethical implications of coercive practices, and how can they be overcome? {{RoundBoxBottom}} == Therapeutic relationships == {{ic|Include an introductory paragraph before branching into sub-sections}} === Defining therapeutic relationships === [[File:Health professional in scrubs enjoying coffee while using a smartphone in a casual work setting during morning hours.jpg|thumb|225px|'''Figure 2.''' Coercive practices can be subtle and hard to spot.]] Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland & Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018) (see Figure 2). Examples of therapeutic relationships in mental health settings include: * Nurses tending to a psychiatric care patient (Ernstmeyer & Christman, 2022). * A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland & Torrico, 2024). === Applying self-determination theory to therapeutic relationships === [[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci & Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan & Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012). * Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider. * Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle. *Relatedness: You feel understood and cared for by your mental healthcare provider. When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan & Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance. === Applying tripartite model of working alliance to therapeutic relationships === The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson & Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward. * Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties. * Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. * Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes. == Coercive practices in mental health contexts == Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner & Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). === Defining coercive practices === Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames & Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). '''Table 1.''' Examples of coercive practices {| class="wikitable" |+ !Restraint !Type of coercion !Example |- |Physical |Formal/explicit |Use of physical force to restrain a patient |- |Mechanical |Formal/explicit |Use of physical tools to restrain a patient |- |Chemical |Formal/explicit |Use of medication such as sedatives to control a patient's behaviour, rather than treating them |- |Environmental |Formal/explicit |Limiting access to environment through practices like seclusion |- |Psychological |Informal/implicit |Using psychological pressures such as implicit threats or consequences to get patients to comply |} <quiz display="simple"> {Coercive practices improve therapeutic relationships: |type="()"} + True - False {Coercive practices are always explicit: |type="()"} - True + False </quiz> === Impact of coercive practices on therapeutic relationships and mental health outcomes === Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes & Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025). '''Table 2.''' Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust {| class="wikitable" style="margin: auto;" |- !Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance |- |Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks. |- |Reduced engagement with mental health care. |Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management. |People may be unable to receive a diagnosis or to set mental health goals. |- |High incidence of trauma/trauma-related disorders, and loss of trust. |Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. |Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. |} === Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes === Coercive practices have negative impacts on those who experience them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016). Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships. Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland & Torrico, 2024) (Sharma & Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025). Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes & Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025). Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma & Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). === Ethical implications of coercive practices === Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner & Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames & Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner & Braun, 2023). == Overcoming coercive practices == Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018). ==Conclusion== In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. ==See also== * [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018) * [[wikipedia:Coercion|Coercion]] (Wikipedia) * [[wikipedia:Motivation|Motivation]] (Wikipedia) * [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia) * [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia) * [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia) * [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia) * [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) * [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia) * [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia) ==References== {{Hanging indent|1= Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., & Alegría, M. (2015). Patient-Provider therapeutic alliance contributes to patient activation in community mental health clinics. Administration and Policy in Mental Health and Mental Health Services Research, 44(4), 431–440. https://doi.org/10.1007/s10488-015-0655-8 Beames, L., & Onwumere, J. (2021). Risk factors associated with use of coercive practices in adult inpatient mental health patients: A systematic review. Journal of Psychiatric and Mental Health Nursing, 29(2). https://doi.org/10.1111/jpm.12757 Carmichael, C., Schiffler, T., Smith, L., Moudatsou, M., Tabaki, I., Doñate-Martínez, A., Alhambra-Borrás, T., Kouvari, M., Karnaki, P., Gil-Salmeron, A., & Grabovac, I. (2023). Barriers and facilitators to health care access for people experiencing homelessness in four european countries: An exploratory qualitative study. International Journal for Equity in Health, 22(1), 206. https://doi.org/10.1186/s12939-023-02011-4 Chieze, M., Clavien, C., Kaiser, S., & Hurst, S. (2021). Coercive measures in psychiatry: A review of ethical arguments. Frontiers in Psychiatry, 12(12). Pubmed Central. https://doi.org/10.3389/fpsyt.2021.790886 Critical Intelligence unit. (2025). Evidence check reducing restrictive practices. https://aci.health.nsw.gov.au/__data/assets/pdf_file/0004/1004377/Evidence-check-Reducing-restrictive-and-coercive-practices.pdf Deci, E. L., & Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27(1), 23–40. https://selfdeterminationtheory.org/SDT/documents/2004_DeciVansteenkiste_SDTandBasicNeedSatisfaction.pdf Ernstmeyer, K., & Christman, E. (2022). Therapeutic communication and the nurse-client relationship. In www.ncbi.nlm.nih.gov. Chippewa Valley Technical College. https://www.ncbi.nlm.nih.gov/books/NBK590036/ (Original work published 2025) Faissner, M., & Braun, E. (2023). The ethics of coercion in mental healthcare: The role of structural racism. Journal of Medical Ethics, 50(7). https://doi.org/10.1136/jme-2023-108984 Gilburt, H., Rose, D., & Slade, M. (2008). The importance of relationships in mental health care: A qualitative study of service users’ experiences of psychiatric hospital admission in the UK. BMC Health Services Research, 8(1). https://doi.org/10.1186/1472-6963-8-92 Gooding, P., Mcsherry, B., Roper, C., & Grey, F. (2018). Alternatives to coercion in mental health settings: A literature review. https://socialequity.unimelb.edu.au/__data/assets/pdf_file/0012/2898525/Alternatives-to-Coercion-Literature-Review-Melbourne-Social-Equity-Institute.pdf Hamovitch, E. K., Choy-Brown, M., & Stanhope, V. (2018). Person-Centered Care and the Therapeutic Alliance. Community Mental Health Journal, 54(7), 951–958. Springer Nature Link. https://doi.org/10.1007/s10597-018-0295-z Hem, M. H., Gjerberg, E., Husum, T. L., & Pedersen, R. (2018). Ethical challenges when using coercion in mental healthcare: A systematic literature review. Nursing Ethics, 25(1), 92–110. Sage Journals. https://doi.org/10.1177/0969733016629770 Iversen, H. W., Riley, H., Råbu, M., & Lorem, G. F. (2025). Building and sustaining therapeutic relationships across treatment settings: A qualitative study of how patients navigate the group dynamics of mental healthcare. BMC Psychiatry, 25(1). https://doi.org/10.1186/s12888-025-06874-5 Johnson, L. N., & Wright, D. W. (2002). Revisiting bordin’s theory on the therapeutic alliance: Implications for family therapy. Contemporary Family Therapy, 24(2), 257–269. https://doi.org/10.1023/a:1015395223978 Kornhaber, R., Walsh, K., Duff, J., & Walker, K. (2016). Enhancing adult therapeutic interpersonal relationships in the acute health care setting: An integrative review. Journal of Multidisciplinary Healthcare, 9(14), 537–546. Pubmed Central. https://doi.org/10.2147/JMDH.S116957 Newton-Howes, G., & Mullen, R. (2011). Coercion in psychiatric care: Systematic review of correlates and themes. Psychiatric Services, 62(5), 465–470. https://doi.org/10.1176/ps.62.5.pss6205_0465 Ng, J. Y. Y., Ntoumanis, N., Thøgersen-Ntoumani, C., Deci, E. L., Ryan, R. M., Duda, J. L., & Williams, G. C. (2012). Self-Determination theory applied to health contexts. Perspectives on Psychological Science, 7(4), 325–340. https://doi.org/10.1177/1745691612447309 Norcross, J. C. (2010). The therapeutic relationship. Psycnet.apa.org; American Psychological Association. https://doi.org/10.1037/12075-004 Opland, C., & Torrico, T. J. (2024, October 6). Psychotherapy and therapeutic relationship. National Library of Medicine; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK608012/ Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68 Ryan, R. M., & Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753 Sashidharan, S. P., Mezzina, R., & Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350 Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version Sharma, N., & Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/ Stanhope, V., Marcus, S., & Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183 Wostry, F., Hahn, S., & Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012 }} ==External links== * [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com) * [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Control]] [[Category:Motivation and emotion/Book/Psychotherapy]] [[Category:Motivation and emotion/Book/Therapeutic alliance]] igvfoakaq3hhbbewy3xvx3v783xlc75 2816022 2816021 2026-06-16T21:56:22Z Jtneill 10242 removed [[Category:Motivation and emotion/Book/Psychotherapy]] using [[Help:Gadget-HotCat|HotCat]] 2816022 wikitext text/x-wiki {{title|Coercion and therapeutic alliance:<br>How do coercive practices in mental health care undermine trust and therapeutic relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} ;Scenario [[File:Healthcare professional writing.jpg|thumb|225px|'''Figure 1'''.Patients may feel pressured into complying by healthcare providers.]] Sarah has struggled with major depressive disorder for much of her life and has been in and out of psychiatric care since she was a teenager. While Sarah is seeking out mental health care by her own choice, she has experienced involuntary admission several times in the past. When she was in psychiatric care, she was often physically restrained due to staff concerns about her own safety. She was also frequently referred to as a "difficult patient". Because of this, Sarah struggles to feel secure in mental health settings. She finds it hard to trust practitioners, and often feels judged due to her history of re-hospitalisations. After unsuccessful experiences with several other therapists, and being on a waiting list for several months, Sarah finally has her first appointment with a new therapist. However, in her first session, Sarah struggles to connect with and feel heard by her therapist. Instead of discussing Sarah's diagnosis and treatment goals, her therapist looks at her mental healthcare record and gives Sarah an impersonal treatment plan that does not align with her needs. Although Sarah is unhappy with the treatment plan, she struggles to voice her concerns because she is worried about being a difficult patient. After her appointment, Sarah feels dehumanised and struggles to feel confident about improving her mental health at all. She wonders if she should even bother returning to her new therapist for a follow-up appointment, or whether she should give up on trying to find mental health support at all. Sarah has experienced some common forms of coercive practice which are pervasive in mental health settings (see Figure 1). * What coercive practices has Sarah experienced? * What factors put Sarah at a higher risk of experiencing coercive practices? * What barriers to good therapeutic relationships are experienced by Sarah? * What could be done differently do improve her treatment outcomes? {{RoundBoxBottom}} {{expand}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What are therapeutic relationships and coercive practices? * How can self-determination theory and the tripartite model of working alliance be applied to understand therapeutic relationships? * What are the impacts of coercive practices on therapeutic relationships and mental health outcomes? * What factors can make someone vulnerable to coercive practices? * What are the ethical implications of coercive practices, and how can they be overcome? {{RoundBoxBottom}} == Therapeutic relationships == {{ic|Include an introductory paragraph before branching into sub-sections}} === Defining therapeutic relationships === [[File:Health professional in scrubs enjoying coffee while using a smartphone in a casual work setting during morning hours.jpg|thumb|225px|'''Figure 2.''' Coercive practices can be subtle and hard to spot.]] Therapeutic relationships are a cornerstone of mental healthcare, {{g}} a [[wikipedia:Therapeutic_relationship|therapeutic relationship]] ([[wikipedia:Therapeutic_alliance|therapeutic alliance]]) is the professional relationship between a mental healthcare provider and a patient (Norcross, 2010). Therapeutic relationships are built on mutual trust, collaboration, empathy and respect (often referred to as [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport rapport])(Opland & Torrico, 2024).Therapeutic relationships are essential for various forms of mental health support, including: {{g}} child and adult [[wikipedia:Psychotherapy|psychotherapy]], [[wikipedia:Psychiatry|psychiatric care]], [[wikipedia:Substance_abuse|substance abuse]], [[wikipedia:Suicidal_ideation|suicidal ideation]] and [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth telehealth] services (Hamovitch et al., 2018). They are also important for patient-centred care, which is treatment that considers an individual's needs and circumstances when creating treatment goals (Hamovitch et al., 2018) (see Figure 2). Examples of therapeutic relationships in mental health settings include: * Nurses tending to a psychiatric care patient (Ernstmeyer & Christman, 2022). * A therapist or councillor woking with a patient/client to improve mental health outcomes (Opland & Torrico, 2024). === Applying self-determination theory to therapeutic relationships === [[wikipedia:Self-determination_theory|Self-determination theory]] (SDT) is a theory of [[wikipedia:Motivation|motivation]] that suggests that people are inherently motivated to grow and develop, as well as master internal and external forces they encounter (Deci & Vansteenkiste, 2004). It also posits that people become intrinsically motivated when psychological needs of autonomy, competence and relatedness are all met (Ryan & Deci, 2000). The components of this theory of motivation can be applied within the context of mental health care and therapeutic relationships. Ng et al., 2012). * Autonomy: Being able to actively participate in the creation of your treatment plan, and feeling free to make choices without being pressured by your mental healthcare provider. * Competence: You feel that you are able to master coping skills or manage mental health symptoms in environments where you would usually struggle. *Relatedness: You feel understood and cared for by your mental healthcare provider. When the psychological needs presented in STD are met in a mental health setting, they contribute to good quality therapeutic relationships, and positive attainment and maintenance of mental health outcomes for patients (Ryan & Deci, 2008). SDT can be used to complement understanding of important factors that contribute to the therapeutic alliance. === Applying tripartite model of working alliance to therapeutic relationships === The tripartite model of working alliance proposes that all therapeutic relationships are composed of three elements, {{g}} the relationship between the therapist and the patient (bond), treatment outcomes that are agreed upon by the patient and therapist (goals), and mutually agreed upon steps to achieve those patient outcomes (tasks) (Johnson & Wright, 2002). The quality of therapeutic relationships can be measured using the working alliance inventory (WAI) (Allen et al., 2015). Applying this model to mental health settings is straightforward. * Bond: You have a strong professional bond with your therapist, there are mutual feelings of trust and care between both parties. * Goals: You and your therapist have discussed and understood your current diagnosis, and what your desired treatment outcomes are. * Tasks: Your treatment plan includes therapy, medications, or techniques that you and your therapist have mutually decided would be the most helpful to achieve your desired treatment outcomes. Much like self-determination theory, the tripartite model of working alliance highlights components that are needed to ensure a good therapeutic relationship, and positive long-lasting treatment outcomes. If these components are neglected, this can damage the therapeutic relationship and undermine treatment outcomes. == Coercive practices in mental health contexts == Despite the known{{f}} importance of good therapeutic relationships in mental health care, coercive practices remain a common occurrence in mental healthcare settings (Sashidharan et al., 2019). The presence of coercive practices in mental health settings revolves around ethical debate over patient autonomy during periods of mental distress (Faissner & Braun, 2023), as well as the potential costs and benefits of coercive measures for patient outcomes (Hem et al., 2018). === Defining coercive practices === Coercive practices refer to any practice that goes against the wishes of a patient/client (Chieze et al., 2021). Coercive practices can be categorised into formal and informal [[wikipedia:Coercion|coercion]] (Chieze et al., 2021). Within these two categories, coercive practices can be further divided into five main types of coercive practices: physical, mechanical, chemical, environmental, and psychological restraint (Beames & Onwumere, 2021). Formal coercion involves explicitly obvious coercive practices, such as limiting a patients{{g}} freedom of movement through practices like seclusion (Chieze et al., 2021). Informal coercion includes any influence or pressure on patient/client decisions, and is often implicit, such as fear of consequences (Chieze et al., 2021). '''Table 1.''' Examples of coercive practices {| class="wikitable" |+ !Restraint !Type of coercion !Example |- |Physical |Formal/explicit |Use of physical force to restrain a patient |- |Mechanical |Formal/explicit |Use of physical tools to restrain a patient |- |Chemical |Formal/explicit |Use of medication such as sedatives to control a patient's behaviour, rather than treating them |- |Environmental |Formal/explicit |Limiting access to environment through practices like seclusion |- |Psychological |Informal/implicit |Using psychological pressures such as implicit threats or consequences to get patients to comply |} <quiz display="simple"> {Coercive practices improve therapeutic relationships: |type="()"} + True - False {Coercive practices are always explicit: |type="()"} - True + False </quiz> === Impact of coercive practices on therapeutic relationships and mental health outcomes === Despite being a common occurrence in mental health settings{{f}}, coercive practices can have detrimental impacts on therapeutic relationships and treatment outcomes (Wostry et al., 2025). Coercive practices such as psychological restraint can make patients feel dehumanised and unheard, and make them feel like they do not have a say in treatment decisions (Newton-Howes & Mullen, 2011). Because of this, patients may be less likely to engage in therapeutic interventions or comply with treatment in the long-term (Wostry et al., 2025). Coercive practices damage trust in health care practitioners and can be traumatic, putting some patients at risk of experiencing [[wikipedia:Post-traumatic_stress_disorder|post-traumatic stress disorder]]. Patients are likely to remember negative experiences caused by coercive practices, which may make them less likely to seek out or engage with necessary mental healthcare in the future (Stanhope et al., 2009). If people do choose to seek out mental healthcare again, they may have trouble sharing their experiences or other relevant information with practitioners (Gilburt et al., 2008). This can lead to worse mental health outcomes related to lack of treatment, or treatment that does not successfully cater to individual needs (Wostry et al., 2025). '''Table 2.''' Applying self-determination theory and the tripartite model of working alliance to highlight how coercive practices undermine therapeutic relationships and trust {| class="wikitable" style="margin: auto;" |- !Coercive practice outcome !! Self determination theory !! Tripartite model of working alliance |- |Reduced patient autonomy. || Autonomy of the patient is not fostered and the need for autonomy is not met, reducing motivation to engage with therapy. ||Patients are not able to collaborate with the practitioner to set treatment tasks. |- |Reduced engagement with mental health care. |Competence need is not fulfilled because people cannot access resources and assistance that would lead to mastery of coping skills or symptom management. |People may be unable to receive a diagnosis or to set mental health goals. |- |High incidence of trauma/trauma-related disorders, and loss of trust. |Relatedness between practitioners and patients cannot be achieved, if it is undermined by negative interactions that do not foster feelings of safety, care and trust. |Traumatic/distressing experiences hinder the possibility of mutual trusting bonds between patients and practitioners. |} === Factors affecting the impact of coercive practices on therapeutic relationships and mental health outcomes === Coercive practices have negative impacts on those who experience them{{f}}. However, some people are more likely have their therapeutic relationships and mental health outcomes undermined by the use of coercive practices{{f}}. Factors that increase this risk can be grouped into three main categories including, client/patient factors, care provider factors, and systemic factors (Sass-Stańczak, 2016; Kornhaber et al., 2016). Vulnerable individuals are more likely to face challenges forming and navigating therapeutic relationships. Vulnerable individuals include people with a serious mental illness, people dealing with substance abuse, people who are experiencing unemployment, and people who face social marginalisation (Iversen et al., 2025). People within these groups are more likely to experience various forms of formal coercion, including involuntary admission, restraint and seclusion (Critical Intelligence unit, 2025). These forms of coercion strip patients of autonomy and damage the foundation of any therapeutic relationships that could have otherwise been formed with practitioners. Because poor therapeutic relationships are associated with less successful treatment outcomes{{f}}, patients may be at a higher risk of re-hospitalisation (Iversen et al., 2025). This can not only erode patient trust in practitioners, but patients with a larger record of hospitalisations are more likely to face stigma from providers (Iversen et al., 2025). This can create a dangerous cycle where vulnerable individuals have poor therapeutic relationships so they receive ineffective care, which puts them at risk of re-hospitalisation. When they re-enter the mental healthcare system, they have prior negative experiences that create more barriers to good therapeutic relationships. Additional client/patient factors also have the potential to impact therapeutic relationships, if these factors are not accommodated sufficiently, therapeutic relationships are at a higher risk of breaking down, and coercive practices are more likely to be used (Sass-Stańczak, 2016). People who face communication challenges due to language/cultural barriers, low education level, or intellectual and developmental disabilities may face challenges when discussing diagnosis and treatment, especially when trying to communicate their needs (Opland & Torrico, 2024) (Sharma & Gupta, 2023). If adequate support is not provided, two way communication between patients and providers can break down (Critical Intelligence unit, 2025). This can result in miscommunication of treatment, and diagnosis and reduce patient's feelings of trust for practitioners, hindering the formation of therapeutic bonds (Critical Intelligence unit, 2025). Mental healthcare providers also have a part to play in ensuring that coercive practices are not used. To ensure a strong therapeutic relationship and avoid use of coercive practices, it is important for mental healthcare providers to cultivate a communication style that facilitates patient feedback and contribution to the therapeutic process (Allen et al., 2015). This is important to avoid patients feeling like they are not heard or respected by practitioners (Newton-Howes & Mullen, 2011). Intervention type is also important, {{g}} interventions involving psycho-education improve patient understanding and assist in building trust, confidence and autonomy (Chieze et al., 2021). Impersonal treatment may undermine these elements of the therapeutic relationship through the use of informal coercion, such as implied treat of not complying to a treatment regimen (Chieze et al., 2021). Practitioners who prioritise patient-centred care over other care approaches are more likely to establish good therapeutic relationships with their clients (Kornhaber et al., 2016). This is because they are able to tailor diagnosis and treatment based on individual needs, rather than attempting to use existing regimens which may be insufficient to diagnose or treat complex long-term issues (Iversen et al., 2025). Systemic factors like issues of short-staffing contributing to provider fatigue in addition to other systemic limitations can reduce access to personalised care, hindering therapeutic relationships (Iversen et al., 2025). Providers who are overworked are less likely to have the emotional and cognitive resources needed to establish and maintain a strong therapeutic relationship with their patients, increasing the risk that coercive measures will be used (Sharma & Gupta, 2023). Additionally, long waiting times due to these systemic issues may cause people to disengage from seeking mental health care, and prevent them from forming beneficial therapeutic relationships and achieving successful mental health outcomes (Carmichael et al., 2023). === Ethical implications of coercive practices === Because coercive practices undermine client/patient autonomy, there is a complex ethical debate over whether their use can be justified in mental health contexts{{f}}. Most of the time, coercive practices are justified when a person is deemed to be a danger to themselves or others, such that their autonomy must be limited for a period of time (Faissner & Braun, 2023). Adult patients admitted into psychiatric care are among the most likely groups to experience mechanical, physical and environmental restraint using this justification (Beames & Onwumere, 2021). Coercive practices relating to involuntary treatment are also justified in cases where an individual is not considered to have the capacity to make autonomous decisions about their own treatment (Hem et al., 2018). These justifications of coercive practices are limited because judgments about safety and a patient's capacity for autonomy are often highly context dependant and subject to the subjective judgment of mental healthcare providers (Hem et al., 2018). Factors such as racism can increase the use of coercive practices against certain groups based on bias such as people of colour (Faissner & Braun, 2023). == Overcoming coercive practices == Although some still argue for the use of coercive practices{{f}}, sighting{{sp}} that the benefits of these practices outweigh the potential damage;{{g}} many mental health care systems and providers recognise the considerable damage caused by coercive practices (Hem et al., 2018. As a result, many practitioners instead advocate for alternative practices that are beneficial to therapeutic relationships and patient outcomes (Hem et al., 2018). Stepping away from coercive practices requires systemic changes that support staff, for example {{g}} staff training that improves risk assessment skills or behaviour modification techniques can help reduce the use of restraining practices (Critical Intelligence unit, 2025). Similarly, equipping staff to provide trauma informed care is also effective at reducing coercive practices, which may be especially relevant if a patient has had traumatic experiences in the mental healthcare system previously (Critical Intelligence unit, 2025). On a community scale, the availability of crisis services can help deescalate potentially harmful situations experienced by vulnerable individuals, reducing the risk of involuntary admission (Gooding et al., 2018). It is also important for healthcare services to update policies and workplace cultures to support staff use of non-coercive practices (Gooding et al., 2018). Finally, it is also essential for mental healthcare practitioners and organisations to not devalue patient's capacity for autonomy despite mental health issues (Hem et al., 2018). ==Conclusion== In mental health care {{g}} strong therapeutic relationships founded in trust, respect and empathy are essential for successful treatment outcomes. However, therapeutic relationships are often undermined by the use of coercive practices which are pervasive in mental healthcare settings. Coercive practices encompass a wide range of practices that erode patient autonomy and can range form explicit forms of restriction such as seclusion or implicit forms of psychological pressure. The impacts of coercive practices can be understood through the lens of self-determination theory and the tripartite model of working alliance. Coercive practices undermine patient autonomy, preventing them from having a say in their treatment outcomes, and reducing motivation to engage in the therapeutic process. Coercive practices also decrease chances that patients will seek out beneficial mental health care, preventing people from setting treatment goals and feeling capable of managing symptoms or improving their mental health. Lastly, coercive practices also increase the risk of patients having traumatic experiences in mental health settings, hindering them from forming important therapeutic bonds with practitioners. Specific factors increase the likelihood of coercive practices, and leaves vulnerable individuals more susceptible to their impacts. People with complex mental health or sociological backgrounds are at higher risk of poor therapeutic relationships and coercive practices. Care provider and systemic factors such as communication style and staffing limitations also play an important role in this regard. While the ethics of coercive practices are debated, many practitioners are moving towards non-coercive practices that foster patient autonomy and support therapeutic relationships such as patient-centred care and tailored treatment. Ultimately, coercive practices are easily relied upon when mental healthcare practitioners are not able to effectively create open and collaborative therapeutic relationships with their patients. To overcome this, mental healthcare settings need to support staff by creating environments where unique patient needs can be accommodated. To achieve this, the review and updating of organisational policy, and staff support are essential. Future research should aim to identify actionable changes that can be made to facilitate these important changes. ==See also== * [[Motivation and emotion/Book/2018/Betrayal and emotion|Betrayal and emotion]] (Book chapter, 2018) * [[wikipedia:Coercion|Coercion]] (Wikipedia) * [[wikipedia:Motivation|Motivation]] (Wikipedia) * [[wikipedia:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (Wikipedia) * [[wikipedia:Psychotherapy|Psychotherapy]] (Wikipedia) * [[wikipedia:Psychiatry|Psychiatry]] (Wikipedia) * [[wikipedia:Substance_abuse|Substance abuse]](Wikipedia) * [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) * [[wikipedia:Suicidal_ideation|Suicidal ideation]] (Wikipedia)* [[wikipedia:Therapeutic_alliance|Therapeutic alliance]] (Wikipedia) * [[wikipedia:Therapeutic_relationship|Therapeutic relationship]] (Wikipedia) ==References== {{Hanging indent|1= Allen, M. L., Cook, B. L., Carson, N., Interian, A., La Roche, M., & Alegría, M. (2015). Patient-Provider therapeutic alliance contributes to patient activation in community mental health clinics. Administration and Policy in Mental Health and Mental Health Services Research, 44(4), 431–440. https://doi.org/10.1007/s10488-015-0655-8 Beames, L., & Onwumere, J. (2021). Risk factors associated with use of coercive practices in adult inpatient mental health patients: A systematic review. Journal of Psychiatric and Mental Health Nursing, 29(2). https://doi.org/10.1111/jpm.12757 Carmichael, C., Schiffler, T., Smith, L., Moudatsou, M., Tabaki, I., Doñate-Martínez, A., Alhambra-Borrás, T., Kouvari, M., Karnaki, P., Gil-Salmeron, A., & Grabovac, I. (2023). Barriers and facilitators to health care access for people experiencing homelessness in four european countries: An exploratory qualitative study. International Journal for Equity in Health, 22(1), 206. https://doi.org/10.1186/s12939-023-02011-4 Chieze, M., Clavien, C., Kaiser, S., & Hurst, S. (2021). Coercive measures in psychiatry: A review of ethical arguments. Frontiers in Psychiatry, 12(12). Pubmed Central. https://doi.org/10.3389/fpsyt.2021.790886 Critical Intelligence unit. (2025). Evidence check reducing restrictive practices. https://aci.health.nsw.gov.au/__data/assets/pdf_file/0004/1004377/Evidence-check-Reducing-restrictive-and-coercive-practices.pdf Deci, E. L., & Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27(1), 23–40. https://selfdeterminationtheory.org/SDT/documents/2004_DeciVansteenkiste_SDTandBasicNeedSatisfaction.pdf Ernstmeyer, K., & Christman, E. (2022). Therapeutic communication and the nurse-client relationship. In www.ncbi.nlm.nih.gov. Chippewa Valley Technical College. https://www.ncbi.nlm.nih.gov/books/NBK590036/ (Original work published 2025) Faissner, M., & Braun, E. (2023). The ethics of coercion in mental healthcare: The role of structural racism. Journal of Medical Ethics, 50(7). https://doi.org/10.1136/jme-2023-108984 Gilburt, H., Rose, D., & Slade, M. (2008). The importance of relationships in mental health care: A qualitative study of service users’ experiences of psychiatric hospital admission in the UK. BMC Health Services Research, 8(1). https://doi.org/10.1186/1472-6963-8-92 Gooding, P., Mcsherry, B., Roper, C., & Grey, F. (2018). Alternatives to coercion in mental health settings: A literature review. https://socialequity.unimelb.edu.au/__data/assets/pdf_file/0012/2898525/Alternatives-to-Coercion-Literature-Review-Melbourne-Social-Equity-Institute.pdf Hamovitch, E. K., Choy-Brown, M., & Stanhope, V. (2018). Person-Centered Care and the Therapeutic Alliance. Community Mental Health Journal, 54(7), 951–958. Springer Nature Link. https://doi.org/10.1007/s10597-018-0295-z Hem, M. H., Gjerberg, E., Husum, T. L., & Pedersen, R. (2018). Ethical challenges when using coercion in mental healthcare: A systematic literature review. Nursing Ethics, 25(1), 92–110. Sage Journals. https://doi.org/10.1177/0969733016629770 Iversen, H. W., Riley, H., Råbu, M., & Lorem, G. F. (2025). Building and sustaining therapeutic relationships across treatment settings: A qualitative study of how patients navigate the group dynamics of mental healthcare. BMC Psychiatry, 25(1). https://doi.org/10.1186/s12888-025-06874-5 Johnson, L. N., & Wright, D. W. (2002). Revisiting bordin’s theory on the therapeutic alliance: Implications for family therapy. Contemporary Family Therapy, 24(2), 257–269. https://doi.org/10.1023/a:1015395223978 Kornhaber, R., Walsh, K., Duff, J., & Walker, K. (2016). Enhancing adult therapeutic interpersonal relationships in the acute health care setting: An integrative review. Journal of Multidisciplinary Healthcare, 9(14), 537–546. Pubmed Central. https://doi.org/10.2147/JMDH.S116957 Newton-Howes, G., & Mullen, R. (2011). Coercion in psychiatric care: Systematic review of correlates and themes. Psychiatric Services, 62(5), 465–470. https://doi.org/10.1176/ps.62.5.pss6205_0465 Ng, J. Y. Y., Ntoumanis, N., Thøgersen-Ntoumani, C., Deci, E. L., Ryan, R. M., Duda, J. L., & Williams, G. C. (2012). Self-Determination theory applied to health contexts. Perspectives on Psychological Science, 7(4), 325–340. https://doi.org/10.1177/1745691612447309 Norcross, J. C. (2010). The therapeutic relationship. Psycnet.apa.org; American Psychological Association. https://doi.org/10.1037/12075-004 Opland, C., & Torrico, T. J. (2024, October 6). Psychotherapy and therapeutic relationship. National Library of Medicine; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK608012/ Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68 Ryan, R. M., & Deci, E. L. (2008). A self-determination theory approach to psychotherapy: The motivational basis for effective change. Canadian Psychology, 49(3), 186–193. https://doi.org/10.1037/a0012753 Sashidharan, S. P., Mezzina, R., & Puras, D. (2019). Reducing coercion in mental healthcare. Epidemiology and Psychiatric Sciences, 28(6), 605–612. https://doi.org/10.1017/s2045796019000350 Sass-Stańczak, K. (2016, January 20). (PDF) therapeutic relationship - what influences it and how does it influence on the psychotherapy process? (english version) (C. Czabala, Ed.). Research Gate; Research Gate. https://www.researchgate.net/publication/291274358_Therapeutic_relationship_-_What_influences_it_and_how_does_it_influence_on_the_psychotherapy_process_english_version Sharma, N., & Gupta, V. (2023). Therapeutic communication. PubMed; StatPearls Publishing. https://www.ncbi.nlm.nih.gov/books/NBK567775/ Stanhope, V., Marcus, S., & Solomon, P. (2009). The impact of coercion on services from the perspective of mental health care consumers with co-occurring disorders. Psychiatric Services, 60(2), 183–188. https://doi.org/10.1176/ps.2009.60.2.183 Wostry, F., Hahn, S., & Schrems, B. (2025). The impact of coercive measures on the therapeutic relationship between patients and nurses in the acute psychiatric care. an integrative review. Journal of Psychiatric and Mental Health Nursing. https://doi.org/10.1111/jpm.70012 }} ==External links== * [https://www.psychologytoday.com/au/blog/insight-therapy/202001/how-therapy-works-the-role-real-rapport How therapy works: the role of real rapport] (Psychologytoday.com) * [https://www.health.gov.au/topics/health-technologies-and-digital-health/about/telehealth Telehealth] (Australian Government: department of health, disability and ageing) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Control]] [[Category:Motivation and emotion/Book/Therapeutic alliance]] bjif2bzb9jxqt9n6xhfhfjijdrh16gr 0 323153 2816019 2815325 2026-06-16T21:54:12Z Jtneill 10242 /* Motivation */ + [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? 2816019 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? 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{{ME-By|User Name}} # [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}} # [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}} # [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}} # [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}} ==Motivation and emotion== # [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}} # [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}} # [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}} # [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}} # [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}} # [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 3k52l0pytp95ihzdi165ypt60q7fyec 2816029 2816019 2026-06-16T22:08:23Z Jtneill 10242 /* Emotion */ [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? 2816029 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}} # [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Building therapeutic alliance/]] - What psychological factors contribute to the development of a strong therapeutic alliance? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}} # [[/Exercise gamification motivation/]] - How can gamification affect exercise motivation and behaviour? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivations for using sex work services/]] - What motivates use of sex work services? {{ME-By|User Name}} # [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Retirement motivation/]] - What motivates retirement from work? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}} # [[/Self-determination theory and dementia care/]] - How can autonomy, competence, and relatedness be supported in people living with dementia? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Volunteer counsellor motivation/]] - What motivates people to become and remain volunteer counsellors? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}} # [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Interpersonal psychotherapy and emotion/]] - How does interpersonal psychotherapy improve emotional wellbeing through changes in relationships? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Irritability/]] - What is irritability, what causes it, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}} # [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}} ==Motivation and emotion== # [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}} # [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}} # [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}} # [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}} # [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}} # [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 0zdmplf3ix2k7k10il6645n219xpvu0 0 326182 2815953 2815918 2026-06-16T14:26:29Z Regliste 3029369 added DOI 2815953 wikitext text/x-wiki {{Article info | last1 = Stiegler | orcid1 = 0009-0001-5789-6923 | first1 = Jean-Baptiste | affiliation1 = Université Paris-Saclay | correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr | journal = WikiJournal of Science | et_al = true | w1 = Pentagram map | from w1 = true | keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems | license = CC-BY-SA 4.0 | submitted = 2025-12-08 | abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992{{Sfn|Schwartz|1992}}. The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}} It admits many generalizations in [[w:Projective space|projective spaces]] and other settings. }} == Introduction == === Informal definition === ==== On polygons ==== [[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]] Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon <math>P</math> with <math>n</math> sides, one can draw the "shortest [[w:diagonal|diagonal]]s", meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new <math>n</math>-gon <math>T(P)</math>; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}} The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image <math>T(P)</math> can fail to be a new <math>n</math>-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}} More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}} ==== On the moduli space of polygons ==== Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and heptagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}} === Historical elements === The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name "[[w:pentagram|pentagram]]".{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}} The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} ==Definitions and first properties== === Definition of the map === [[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]] [[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex <math>w_2</math> is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]] Let <math>n\geq 5</math> be an integer. A polygon <math>P</math> with <math>n</math> sides, or <math>n</math>-gon, is a tuple of [[w:Vertex (geometry)|vertices]] <math>(v_1,\dots,v_n)</math> lying in some [[w:projective plane|projective plane]] <math>\mathbb P ^2</math>,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] <math>n</math>. The [[w:Dimension of an algebraic variety|dimension]] of the space of <math>n</math>-gons is <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.1}} Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive "shortest" [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex <math>v_k</math> and a "second neighbour" <math>v_{k\pm 2}</math>.}} defines a new point<math display="block"> w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. </math>This procedure defines a new <math>n</math>-gon <math>T(P)=(w_1,\dots,w_n)</math>, as in Figure 2.{{Sfn|Schwartz|1992|p=71}} The labeling of the indices of <math>T(P)</math> is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}} The pentagram map on polygons is a [[w:birational map|birational map]] <math>T:(\mathbb P^2)^n</math>{{nowrap|{{font|size=145%|⇢}}}}<math>(\mathbb P^2)^n</math>. Indeed, each [[w:Homogeneous coordinates|coordinate]] of <math>w_k</math> is given as a [[w:rational function|rational function]] of the coordinates of <math>v_{k-1},\dots,v_{k+2}</math>, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections <math>\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} </math>, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}} === Moduli space === The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] <math>\mathbb P \mathrm{GL}_{3}</math>. Such a transformation <math>M</math> [[w:Group action|acts]] on a polygon <math>P</math> by sending it to <math>M \cdot P:=(Mv_1,\dots,Mv_n)</math>. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is <math>2n-8</math>.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} ===Twisted polygons=== [[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]] The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer <math>n\geq5</math>, a twisted <math>n</math>-gon <math>P</math> is the data of: * a [[w:Sequence#Indexing|bi-infinite sequence]] of points <math>(v_k)_{k\in\mathbb Z}</math> in the projective plane (called the vertices), * a [[w:projective transformation|projective transformation]] <math>M \in \mathbb P \mathrm{GL}_3</math> (called the [[w:monodromy|monodromy]]), such that for any <math>k \in \mathbb Z</math>, the property <math>v_{k+n}=Mv_k</math> is satisfied. The dimension of the space of twisted <math>n</math>-gons is <math>2n+8</math>.{{Sfn|Schwartz|2008}} When <math>M=\mathrm{Id}</math>, this gives back the initial definition of polygons (which are said to be closed). The space of closed <math>n</math>-gons is of [[w:codimension|codimension]] <math>8</math> in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}} The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.3}} == Collapsing of convex polygons == === Exponential shrinking === [[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]] Let <math>P</math> be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts. # The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}} # There exists a constant <math>0< \eta_P<1</math>, depending on <math>P</math>, such that for any <math>N \in \mathbb N</math>, the diameters of the iterates verify the inequality <math display="inline">\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). </math>{{Sfn|Schwartz|1992|loc=theorem 3.1}} Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}} The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]]. === Coordinates of the limit point === The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of <math>\mathbb R^3</math>.{{sfn|Glick|2020}} This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon <math>P</math> into a family of twisted ones '''<math>(P_z)_{z\in \mathbb C^*}</math>''' with monodromy <math>M_z</math>. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}} <math display="block">\left.\frac{dM_z}{dz}\right|_{z=1}.</math> === Generalization === The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}} == Periodic orbits on the moduli space == For some configurations of closed polygons, the iterate of the pentagram map will send <math>P</math> to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of <math>P</math> is [[w:Periodic orbit|periodic]]. ===Pentagons and hexagons=== [[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}} The pentagram map <math>T</math> is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate <math>T^2</math> is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}} ==== Generalization ==== The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in <math>\mathbb P ^k</math>, for polygons with <math>k+3</math> or <math>2k+2</math> sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}} === Poncelet polygons === A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet <math>n</math>-gon <math>P</math> lying on the [[w:real projective plane|real projective plane]], the polygon <math>T^2(P)</math> is projectively equivalent to <math>P</math>.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when <math>n</math> is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}} However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}} ==Coordinates for the moduli space== The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section. === Corner coordinates === [[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]] Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be : <math> [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. </math> The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios: : <math>x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],</math> : <math>y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].</math> Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences <math>(x_k)_{k \in \mathbb Z}</math> and <math>(y_k)_{k \in \mathbb Z}</math> associated to a twisted <math>n</math>-gon are <math>n</math> periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}} The corner invariants are elements of <math>\mathbb{P}^1\smallsetminus\{0,1,\infty\}</math>, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted <math>n</math>-gons and <math>(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}</math>.{{Sfn|Weinreich|2022|loc=theorem 3.6}} ===ab-coordinates=== There is a second set of coordinates for the moduli space of twisted <math>n</math>-gons defined over a [[w:Field (mathematics)|field]] <math>F</math> satisfying <math>\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)</math>,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that <math>n</math> is not divisible by <math>3</math>.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}} The vertices <math>v_k</math> in the [[w:projective plane|projective plane]] <math>\mathbb P^2(F)</math> can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] <math>V_k</math> in the [[w:affine space|affine space]] <math>F^3</math> so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to <math>1</math>. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}} : <math>V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.</math> This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}} They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}} : <math>x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},</math> : <math>y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.</math> ==Formulas on the moduli space== ===As a birational map === The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}} : <math>x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},</math> : <math>y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.</math> === The scaling symmetry === The [[w:multiplicative group|multiplicative group]] <math>F\smallsetminus\{0\}</math> [[w:One-parameter group|acts]] on the moduli space in the following way: : <math>R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),</math> where <math>R</math> is called the scaling action and <math>s</math> is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}} ==Invariant structures== ===Monodromy invariants=== The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are :<math> O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. </math> The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here. Let <math>M\in \mathrm{GL}_3</math> be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted <math>n</math>-gon. The quantities : <math>\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},</math> are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities : <math>\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,</math> have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} : <math> \tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad \tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3, </math> where each <math>O_k</math> and <math>E_k</math> are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight <math>k</math> and <math>-k</math>,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}} : <math> R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. </math> The quantities <math>O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,</math> are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}} ==== Polygons on conics ==== Whenever <math>P</math> is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has <math>O_k(P)=E_k(P)</math> for all <math>k</math>.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if <math>P</math> is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}} ===Poisson bracket=== An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it: <math display="block"> \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 </math>for all <math>i,j</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} The Poisson bracket is defined in terms of the corner coordinates by: <math display="block"> \begin{align} \{x_i,x_{i\pm1}\} &= \mp x_i x_{i+1}, \\ \{y_i,y_{i\pm 1}\} &= \mp y_i y_{i+1}, \\ \{x_i,x_j\} &= \{y_i,y_j\} = \{x_i,y_j\} = 0 \end{align}</math>for all other <math> i,j.</math>{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}} === The spectral curve === Let <math>\zeta</math> be an element of the [[w:multiplicative group|multiplicative group]] and <math>P_\zeta</math> be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] <math>R_\zeta</math> on <math>P</math>. A [[w:Lax matrix|Lax matrix]] <math>\hat{T}(\zeta) \in \mathrm{GL}_3</math> is a lift of the monodromy of <math>P_\zeta</math> satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]] <math display="block"> Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),</math>or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation <math>Q(\lambda,\zeta)=0</math>.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of <math>Q</math>.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is <math>n-1</math> if <math>n</math> is odd, and <math>n-2</math> if <math>n</math> is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}} It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}} ==Complete integrability== The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} ===Arnold–Liouville integrability=== The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants <math>O_n</math> and <math>E_n</math> are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that<math display="block"> \{O_n,f\}=\{E_n,f\} = 0 </math>for all functions <math>f</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When <math>n</math> is even, this is also true for the monodromy invariants <math>O_{\lfloor n/2 \rfloor }</math> and <math>E_{\lfloor n/2 \rfloor }</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}} Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}} ===Algebro-geometric integrability=== In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}} === Dimension of the invariant manifold === For twisted <math>n</math>-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}} : <math>\begin{cases} n-1 & \text{when }n \text{ is odd,}\\ n-2 & \text{when }n \text{ is even.} \end{cases}</math> Moreover, when <math>n</math> is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}} === For closed polygons === There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}} Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}} In both situation, the dimension of the invariant manifolds decreases by <math>3</math> for closed <math>n</math>-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}} :<math>\begin{cases} n-4 & \text{when }n \text{ is odd,}\\ n-5 & \text{when }n \text{ is even.} \end{cases}</math> ==Connections to other topics== ===The Boussinesq equation=== The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}} Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve <math> C:\mathbb R\to \mathbb R^2 </math> and real numbers <math>x</math> and <math>t</math>, consider the [[w:chord (geometry)|chord]] connecting <math> C(x-t) </math> to <math> C(x+t) </math>. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve <math> C_t(x) </math>. When <math>t</math> is extremely small, the curve <math> C_t(x) </math> is a good model for the time <math>t</math> evolution of the original curve <math> C_0(x) </math> under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}} ===Cluster algebras=== The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. This provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}} === Singularity theory === The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon <math>P</math> is [[w:Singular point of an algebraic variety|singular]] for the pentagram map <math>T</math>, then there exists an integer <math>m</math> such that <math>P</math> not singular for the iterate map <math>T^m</math>.{{Sfn|Glick|2012}} Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon <math>P</math> is singular for some iterate of the pentagram map <math>T^m</math>, then it will also be singular for some iterate of the inverse map <math>T^{-m'}</math>.{{Sfn|Glick|2015}} == Generalizations == The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] <math>\mathbb P^d</math>, under the action of the [[w:Projective linear group|projective group]] <math>\mathbb P \mathrm{GL}_{d+1}</math>. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}} === Polygons in general positions === Let <math>d \geq 2</math> and <math>P</math> be a twisted polygon of <math>\mathbb P^d</math> in [[w:general position|general position]]. ==== Short diagonal pentagram maps ==== The <math>k</math>-th ''short diagonal hyperplane'' <math>H_k^{sh}</math> is uniquely defined by passing through the vertices <math>v_k,v_{k+2},\dots,v_{k+2d-2}</math>. [[w:Generic property#In algebraic geometry|Generically]], the intersection of <math>d</math> consecutive hyperplanes uniquely defines a new point : <math>T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.</math> Doing this for every vertex defines a new twisted polygon. This map, denoted by <math>T_{sh}</math>, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}} ==== Generalized pentagram maps ==== The previous procedure can be generalized. Let <math>I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})</math> be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the <math>k</math>-th hyperplane <math>H_k^I</math> to be passing through the vertices <math>v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}</math>. A new point is given by the intersection : <math>T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.</math> The map <math>T_{I,J}</math> is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering<math>d=2,~I=(2),~J=(1)</math>. Integrability can be numerically tested by picking a random polygon <math>P</math> with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps <math>T_{I,I}</math> are integrable for any <math>I</math>.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}} Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}} ==== Dented pentagram maps ==== Fix an integer <math>m\in \{1,\dots ,d-1\}</math>. Consider the jump tuple <math>I_m:=(1,\dots,1,2,1,\dots,1)</math>, where the <math>2</math> is at the <math>m</math>-th place, and the intersection tuple <math>J:=(1,\dots,1)</math>. The dented pentagram map is <math>T_m :=T_{I_m,J}</math>. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}} For an integer <math>p \geq 2</math>, the deep dented pentagram map (of depth <math>p</math>) <math>T_m^p</math> is the same map as before, but the number <math>2</math> in the definition of <math>I_m</math> is replaced by <math>p</math>. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}} === Corrugated polygons === A twisted polygon <math>P</math> lying in <math>\mathbb P^d</math> is said to be corrugated if for any <math>k\in \mathbb Z</math>, the vertices <math>v_k,v_{k+1},v_{k+d},v_{k+d+1}</math> span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by : <math>T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.</math> The map <math>T_\text{cor}</math> yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}} In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}} === Grassmannian polygons === Let <math>d \geq 3, m \geq 1</math> be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space <math>\mathrm{Gr}(m,md)</math>, which consists of <math>m</math>-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an <math>md</math>-dimensional [[w:vector space|vector space]]. When <math>m=1</math>, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s <math>\mathbb P^d</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} A point <math>v\in\operatorname{Gr}(m,md)</math> is represented by an <math>md \times m</math> matrix <math>X_v</math> such that its columns form a [[w:Basis (linear algebra)|basis]] of <math>v</math>. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] <math>\mathrm{GL}_{md}</math> by multiplication on the left of <math>X_v</math>. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for <math>v</math>, and because some matrices act trivially.}} Hence, the polygons of <math>\mathrm{Gr}(m,md)</math> and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} Depending on the parity of <math>d</math>, one can define linear subspaces spanned by some <math>X_{v_k}</math>'s such that taking their intersection generically defines a new point <math>v\in\mathrm{Gr}(m,md)</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}} === Over rings === The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmanians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}} == References == {{reflist|25em}} ===Notes=== {{notelist}} ==Works cited== *{{Cite journal |ref=harv |title=The Limit Point of the Pentagram Map and Infinitesimal Monodromy |url=https://academic.oup.com/imrn/article/2022/7/5383/5911460 |journal=International Mathematics Research Notices |date=2022-03-23 |issn=1073-7928 |pages=5383–5397 |volume=2022 |issue=7 |doi=10.1093/imrn/rnaa258 |language=en |first1=Quinton |last1=Aboud |first2=Anton |last2=Izosimov}} *{{Cite journal|ref=harv |title=Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder|journal=Symmetry, Integrability and Geometry: Methods and Applications|date=2025-06-03|issn=1815-0659|doi=10.3842/sigma.2025.040|first1=Niklas Christoph|last1=Affolter|first2=Terrence|last2=George|first3=Sanjay|last3=Ramassamy}} *{{Cite journal |ref=harv |last=Berger |first=Marcel |author-link=w:Marcel Berger 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jean-baptiste.stiegler@universite-paris-saclay.fr | journal = WikiJournal of Science | et_al = true | w1 = Pentagram map | from w1 = true | keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems | license = CC-BY-SA 4.0 | submitted = 2025-12-08 | abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}} The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}} It admits many generalizations in [[w:Projective space|projective spaces]] and other settings. }} == Introduction == === Informal definition === ==== On polygons ==== [[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]] Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon <math>P</math> with <math>n</math> sides, one can draw the "shortest [[w:diagonal|diagonal]]s", meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new <math>n</math>-gon <math>T(P)</math>; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}} The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image <math>T(P)</math> can fail to be a new <math>n</math>-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}} More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}} ==== On the moduli space of polygons ==== Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and heptagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}} === Historical elements === The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name "[[w:pentagram|pentagram]]".{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}} The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} ==Definitions and first properties== === Definition of the map === [[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]] [[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex <math>w_2</math> is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]] Let <math>n\geq 5</math> be an integer. A polygon <math>P</math> with <math>n</math> sides, or <math>n</math>-gon, is a tuple of [[w:Vertex (geometry)|vertices]] <math>(v_1,\dots,v_n)</math> lying in some [[w:projective plane|projective plane]] <math>\mathbb P ^2</math>,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] <math>n</math>. The [[w:Dimension of an algebraic variety|dimension]] of the space of <math>n</math>-gons is <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.1}} Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive "shortest" [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex <math>v_k</math> and a "second neighbour" <math>v_{k\pm 2}</math>.}} defines a new point<math display="block"> w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. </math>This procedure defines a new <math>n</math>-gon <math>T(P)=(w_1,\dots,w_n)</math>, as in Figure 2.{{Sfn|Schwartz|1992|p=71}} The labeling of the indices of <math>T(P)</math> is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}} The pentagram map on polygons is a [[w:birational map|birational map]] <math>T:(\mathbb P^2)^n</math>{{nowrap|{{font|size=145%|⇢}}}}<math>(\mathbb P^2)^n</math>. Indeed, each [[w:Homogeneous coordinates|coordinate]] of <math>w_k</math> is given as a [[w:rational function|rational function]] of the coordinates of <math>v_{k-1},\dots,v_{k+2}</math>, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections <math>\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} </math>, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}} === Moduli space === The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] <math>\mathbb P \mathrm{GL}_{3}</math>. Such a transformation <math>M</math> [[w:Group action|acts]] on a polygon <math>P</math> by sending it to <math>M \cdot P:=(Mv_1,\dots,Mv_n)</math>. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is <math>2n-8</math>.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} ===Twisted polygons=== [[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]] The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer <math>n\geq5</math>, a twisted <math>n</math>-gon <math>P</math> is the data of: * a [[w:Sequence#Indexing|bi-infinite sequence]] of points <math>(v_k)_{k\in\mathbb Z}</math> in the projective plane (called the vertices), * a [[w:projective transformation|projective transformation]] <math>M \in \mathbb P \mathrm{GL}_3</math> (called the [[w:monodromy|monodromy]]), such that for any <math>k \in \mathbb Z</math>, the property <math>v_{k+n}=Mv_k</math> is satisfied. The dimension of the space of twisted <math>n</math>-gons is <math>2n+8</math>.{{Sfn|Schwartz|2008}} When <math>M=\mathrm{Id}</math>, this gives back the initial definition of polygons (which are said to be closed). The space of closed <math>n</math>-gons is of [[w:codimension|codimension]] <math>8</math> in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}} The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension <math>2n</math>.{{Sfn|Weinreich|2022|loc=definition 1.3}} == Collapsing of convex polygons == === Exponential shrinking === [[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]] Let <math>P</math> be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts. # The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}} # There exists a constant <math>0< \eta_P<1</math>, depending on <math>P</math>, such that for any <math>N \in \mathbb N</math>, the diameters of the iterates verify the inequality <math display="inline">\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). </math>{{Sfn|Schwartz|1992|loc=theorem 3.1}} Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}} The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]]. === Coordinates of the limit point === The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of <math>\mathbb R^3</math>.{{sfn|Glick|2020}} This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon <math>P</math> into a family of twisted ones '''<math>(P_z)_{z\in \mathbb C^*}</math>''' with monodromy <math>M_z</math>. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}} <math display="block">\left.\frac{dM_z}{dz}\right|_{z=1}.</math> === Generalization === The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}} == Periodic orbits on the moduli space == For some configurations of closed polygons, the iterate of the pentagram map will send <math>P</math> to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of <math>P</math> is [[w:Periodic orbit|periodic]]. ===Pentagons and hexagons=== [[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}} The pentagram map <math>T</math> is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate <math>T^2</math> is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}} ==== Generalization ==== The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in <math>\mathbb P ^k</math>, for polygons with <math>k+3</math> or <math>2k+2</math> sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}} === Poncelet polygons === A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet <math>n</math>-gon <math>P</math> lying on the [[w:real projective plane|real projective plane]], the polygon <math>T^2(P)</math> is projectively equivalent to <math>P</math>.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when <math>n</math> is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}} However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}} ==Coordinates for the moduli space== The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section. === Corner coordinates === [[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]] Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be : <math> [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. </math> The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios: : <math>x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],</math> : <math>y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].</math> Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences <math>(x_k)_{k \in \mathbb Z}</math> and <math>(y_k)_{k \in \mathbb Z}</math> associated to a twisted <math>n</math>-gon are <math>n</math> periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}} The corner invariants are elements of <math>\mathbb{P}^1\smallsetminus\{0,1,\infty\}</math>, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted <math>n</math>-gons and <math>(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}</math>.{{Sfn|Weinreich|2022|loc=theorem 3.6}} ===ab-coordinates=== There is a second set of coordinates for the moduli space of twisted <math>n</math>-gons defined over a [[w:Field (mathematics)|field]] <math>F</math> satisfying <math>\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)</math>,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that <math>n</math> is not divisible by <math>3</math>.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}} The vertices <math>v_k</math> in the [[w:projective plane|projective plane]] <math>\mathbb P^2(F)</math> can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] <math>V_k</math> in the [[w:affine space|affine space]] <math>F^3</math> so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to <math>1</math>. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}} : <math>V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.</math> This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}} They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}} : <math>x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},</math> : <math>y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.</math> ==Formulas on the moduli space== ===As a birational map === The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}} : <math>x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},</math> : <math>y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.</math> === The scaling symmetry === The [[w:multiplicative group|multiplicative group]] <math>F\smallsetminus\{0\}</math> [[w:One-parameter group|acts]] on the moduli space in the following way: : <math>R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),</math> where <math>R</math> is called the scaling action and <math>s</math> is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}} ==Invariant structures== ===Monodromy invariants=== The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are :<math> O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. </math> The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here. Let <math>M\in \mathrm{GL}_3</math> be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted <math>n</math>-gon. The quantities : <math>\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},</math> are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities : <math>\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,</math> have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} : <math> \tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad \tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3, </math> where each <math>O_k</math> and <math>E_k</math> are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight <math>k</math> and <math>-k</math>,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}} : <math> R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. </math> The quantities <math>O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,</math> are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}} ==== Polygons on conics ==== Whenever <math>P</math> is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has <math>O_k(P)=E_k(P)</math> for all <math>k</math>.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if <math>P</math> is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}} ===Poisson bracket=== An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it: <math display="block"> \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 </math>for all <math>i,j</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} The Poisson bracket is defined in terms of the corner coordinates by: <math display="block"> \begin{align} \{x_i,x_{i\pm1}\} &= \mp x_i x_{i+1}, \\ \{y_i,y_{i\pm 1}\} &= \mp y_i y_{i+1}, \\ \{x_i,x_j\} &= \{y_i,y_j\} = \{x_i,y_j\} = 0 \end{align}</math>for all other <math> i,j.</math>{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}} === The spectral curve === Let <math>\zeta</math> be an element of the [[w:multiplicative group|multiplicative group]] and <math>P_\zeta</math> be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] <math>R_\zeta</math> on <math>P</math>. A [[w:Lax matrix|Lax matrix]] <math>\hat{T}(\zeta) \in \mathrm{GL}_3</math> is a lift of the monodromy of <math>P_\zeta</math> satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]] <math display="block"> Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),</math>or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation <math>Q(\lambda,\zeta)=0</math>.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of <math>Q</math>.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is <math>n-1</math> if <math>n</math> is odd, and <math>n-2</math> if <math>n</math> is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}} It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}} ==Complete integrability== The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} ===Arnold–Liouville integrability=== The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants <math>O_n</math> and <math>E_n</math> are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that<math display="block"> \{O_n,f\}=\{E_n,f\} = 0 </math>for all functions <math>f</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When <math>n</math> is even, this is also true for the monodromy invariants <math>O_{\lfloor n/2 \rfloor }</math> and <math>E_{\lfloor n/2 \rfloor }</math>.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}} Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}} ===Algebro-geometric integrability=== In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}} === Dimension of the invariant manifold === For twisted <math>n</math>-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}} : <math>\begin{cases} n-1 & \text{when }n \text{ is odd,}\\ n-2 & \text{when }n \text{ is even.} \end{cases}</math> Moreover, when <math>n</math> is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}} === For closed polygons === There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}} Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}} In both situation, the dimension of the invariant manifolds decreases by <math>3</math> for closed <math>n</math>-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}} :<math>\begin{cases} n-4 & \text{when }n \text{ is odd,}\\ n-5 & \text{when }n \text{ is even.} \end{cases}</math> ==Connections to other topics== ===The Boussinesq equation=== The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}} Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve <math> C:\mathbb R\to \mathbb R^2 </math> and real numbers <math>x</math> and <math>t</math>, consider the [[w:chord (geometry)|chord]] connecting <math> C(x-t) </math> to <math> C(x+t) </math>. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve <math> C_t(x) </math>. When <math>t</math> is extremely small, the curve <math> C_t(x) </math> is a good model for the time <math>t</math> evolution of the original curve <math> C_0(x) </math> under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}} ===Cluster algebras=== The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. This provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}} === Singularity theory === The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon <math>P</math> is [[w:Singular point of an algebraic variety|singular]] for the pentagram map <math>T</math>, then there exists an integer <math>m</math> such that <math>P</math> not singular for the iterate map <math>T^m</math>.{{Sfn|Glick|2012}} Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon <math>P</math> is singular for some iterate of the pentagram map <math>T^m</math>, then it will also be singular for some iterate of the inverse map <math>T^{-m'}</math>.{{Sfn|Glick|2015}} == Generalizations == The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] <math>\mathbb P^d</math>, under the action of the [[w:Projective linear group|projective group]] <math>\mathbb P \mathrm{GL}_{d+1}</math>. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}} === Polygons in general positions === Let <math>d \geq 2</math> and <math>P</math> be a twisted polygon of <math>\mathbb P^d</math> in [[w:general position|general position]]. ==== Short diagonal pentagram maps ==== The <math>k</math>-th ''short diagonal hyperplane'' <math>H_k^{sh}</math> is uniquely defined by passing through the vertices <math>v_k,v_{k+2},\dots,v_{k+2d-2}</math>. [[w:Generic property#In algebraic geometry|Generically]], the intersection of <math>d</math> consecutive hyperplanes uniquely defines a new point : <math>T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.</math> Doing this for every vertex defines a new twisted polygon. This map, denoted by <math>T_{sh}</math>, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}} ==== Generalized pentagram maps ==== The previous procedure can be generalized. Let <math>I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})</math> be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the <math>k</math>-th hyperplane <math>H_k^I</math> to be passing through the vertices <math>v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}</math>. A new point is given by the intersection : <math>T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.</math> The map <math>T_{I,J}</math> is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering<math>d=2,~I=(2),~J=(1)</math>. Integrability can be numerically tested by picking a random polygon <math>P</math> with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps <math>T_{I,I}</math> are integrable for any <math>I</math>.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}} Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}} ==== Dented pentagram maps ==== Fix an integer <math>m\in \{1,\dots ,d-1\}</math>. Consider the jump tuple <math>I_m:=(1,\dots,1,2,1,\dots,1)</math>, where the <math>2</math> is at the <math>m</math>-th place, and the intersection tuple <math>J:=(1,\dots,1)</math>. The dented pentagram map is <math>T_m :=T_{I_m,J}</math>. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}} For an integer <math>p \geq 2</math>, the deep dented pentagram map (of depth <math>p</math>) <math>T_m^p</math> is the same map as before, but the number <math>2</math> in the definition of <math>I_m</math> is replaced by <math>p</math>. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}} === Corrugated polygons === A twisted polygon <math>P</math> lying in <math>\mathbb P^d</math> is said to be corrugated if for any <math>k\in \mathbb Z</math>, the vertices <math>v_k,v_{k+1},v_{k+d},v_{k+d+1}</math> span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by : <math>T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.</math> The map <math>T_\text{cor}</math> yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}} In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}} === Grassmannian polygons === Let <math>d \geq 3, m \geq 1</math> be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space <math>\mathrm{Gr}(m,md)</math>, which consists of <math>m</math>-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an <math>md</math>-dimensional [[w:vector space|vector space]]. When <math>m=1</math>, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s <math>\mathbb P^d</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} A point <math>v\in\operatorname{Gr}(m,md)</math> is represented by an <math>md \times m</math> matrix <math>X_v</math> such that its columns form a [[w:Basis (linear algebra)|basis]] of <math>v</math>. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] <math>\mathrm{GL}_{md}</math> by multiplication on the left of <math>X_v</math>. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for <math>v</math>, and because some matrices act trivially.}} Hence, the polygons of <math>\mathrm{Gr}(m,md)</math> and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}} Depending on the parity of <math>d</math>, one can define linear subspaces spanned by some <math>X_{v_k}</math>'s such that taking their intersection generically defines a new point <math>v\in\mathrm{Gr}(m,md)</math>.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}} === Over rings === The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmanians. 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2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math></math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>r_{8}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} pkunosvl6tf6huo9zohb2f8rvkrado1 2815940 2815936 2026-06-16T13:48:25Z Dc.samizdat 2856930 /* The 600-cell */ 2815940 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>r_7</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>r_{8}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} pq1nsbhlz5vciteo78ozsjaa240re61 2815942 2815940 2026-06-16T13:51:46Z Dc.samizdat 2856930 /* The 600-cell */ 2815942 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math></math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |95.5~° | rowspan="3" |<math>r_{8}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{7}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 5rmaw03q25ogvs8ycxx30sm5g19jk9a 2815944 2815942 2026-06-16T13:57:28Z Dc.samizdat 2856930 2815944 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{7}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} d9p7iw6jleh76lh75glnum37ymf1wbv 2815945 2815944 2026-06-16T14:08:27Z Dc.samizdat 2856930 /* The 600-cell */ 2815945 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} i6fn17fhc4wpg5ehcqb6pi5cj3z669w 2815947 2815945 2026-06-16T14:15:37Z Dc.samizdat 2856930 /* The 600-cell */ 2815947 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} k1r3s8n0j73da31w4n2fbz05mvf7p3q 2815949 2815947 2026-06-16T14:16:09Z Dc.samizdat 2856930 /* The 600-cell */ 2815949 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} s0hi1r5z0wmzp8ewyekvqm8wcpoh51l 2815954 2815949 2026-06-16T14:26:54Z Dc.samizdat 2856930 /* The 600-cell */ 2815954 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>1/\phi</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} p9vtggatv242l51ki4tsaj7r5xss5vx 2815955 2815954 2026-06-16T14:29:55Z Dc.samizdat 2856930 /* The 600-cell */ 2815955 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. Four distinct pairs of chord lengths are each identified with a pair of congruent [[w:600-cell#Polyhedral sections|polyhedral sections of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section of radius <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. At radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius <math>c_1</math> is a tetrahedron vertex figure, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell. At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance, and its base is the polyhedral section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 7eofsqhat9nqzt7t3bpq9lq376icb7x 2815978 2815955 2026-06-16T16:53:09Z Dc.samizdat 2856930 /* The 600-cell */ 2815978 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with similar effects on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with similar effects on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with similar effects on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} b6lozz9rtf1pnmo4s6l0y638ybbln3w 2815979 2815978 2026-06-16T17:02:26Z Dc.samizdat 2856930 2815979 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. 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-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 823han520wo3rdyocbkub6ajvj1o91b 2815980 2815979 2026-06-16T17:15:51Z Dc.samizdat 2856930 /* Hypercubes */ 2815980 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} tv4sgv6xkij0qlyh1mhpggb57zvj76z 2815981 2815980 2026-06-16T17:28:59Z Dc.samizdat 2856930 /* The 24-cell */ 2815981 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 17h655btzeb9aq8tkpq8ds7m86h57uj 2815982 2815981 2026-06-16T17:30:14Z Dc.samizdat 2856930 /* Hypercubes */ 2815982 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == The 5-cell 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} n0nvwjdy4pwoochbukd7ry7kh69w3ji 2815983 2815982 2026-06-16T17:32:57Z Dc.samizdat 2856930 2815983 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} epn0cxyn3edqg7zjy8281rr8s6h7rtf 2815984 2815983 2026-06-16T17:36:47Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815984 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol {5,3,3}. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} bjmqyexslogdl3w4lgneb5uonhn6cqx 2815985 2815984 2026-06-16T17:37:38Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815985 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... The list of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 0tca10z75zv9my705fnv56l42b9m2y5 2815986 2815985 2026-06-16T17:45:29Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815986 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. The list of 30 chords <math>c_{t}</math> in the 120-cell can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 7gzbijgy0vihn47txo6aojjttjixyd5 2815987 2815986 2026-06-16T17:46:39Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815987 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} | | |{{radic|2.618~}} | | |- style="background: yellow;" | |1.176~ | | |1.618~ | | |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | | | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} | | |{{radic|2.5}} | | |- style="background: palegreen;" | |1.224~ | | |1.581~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} | | |{{radic|2.309~}} | | |- style="background: gainsboro;" | |1.300~ | | |1.520~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | | | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | | | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} | | |{{radic|2.191~}} | | |- style="background: gainsboro;" | |1.345~ | | |1.480~ | | |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | | | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} | | |{{radic|2}} | | |- style="background: seashell;" | |1.414~ | | |1.414~ | | |} The list of 30 chords <math>c_{t}</math> in the 120-cell can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} o9rhrkq8s6l58ddddg1a9sv98sotogt 2815988 2815987 2026-06-16T18:00:27Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815988 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 30 chords <math>c_{t}</math> in the 120-cell can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. All those polytopes except the 5-cell occur in the 600-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 6wpz0h16w5apozykez1yd01kkkpfupk 2815989 2815988 2026-06-16T18:06:02Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815989 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 8fr986c397llqifr47mro21ygfwfax6 2815990 2815989 2026-06-16T18:06:30Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815990 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]] |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px|{30/1}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]] |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]] |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]] |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]] |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]] |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]] |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]] |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]] |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} dujinihshczh62hc4l782o76rav14z8 2815991 2815990 2026-06-16T18:09:34Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2815991 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row uniquely identifies the discrete isoclinic rotation of the 600-cell in invariant central planes containing edges of the short chord {30}-gon. The long chord {30}-gon is the isoclinic rotation's Clifford polygon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} g7m6mq4hqwny162xkdobs5dod1ta374 2816012 2815991 2026-06-16T19:41:10Z Dc.samizdat 2856930 /* The 600-cell */ 2816012 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 1bip2wa08paojgcnleymy40mhslxxw9 2816014 2816012 2026-06-16T19:55:01Z Dc.samizdat 2856930 2816014 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} f12nnoz08jzha6mzye1ncmztlwl9g8r 2816015 2816014 2026-06-16T20:01:38Z Dc.samizdat 2856930 /* The 600-cell */ 2816015 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each row is also identified non-uniquely with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 7anx7tyws88oc7zlu6xt2jqve0k91hs 2816017 2816015 2026-06-16T20:10:13Z Dc.samizdat 2856930 /* The 600-cell */ 2816017 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 07cxh9d7k6pjyi7py73e8lisllxsdj5 2816057 2816017 2026-06-17T05:08:21Z Dc.samizdat 2856930 /* The 600-cell */ 2816057 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} cq1zmya60jrbid9hxjnjlvdgj2q9ib8 2816058 2816057 2026-06-17T05:12:06Z Dc.samizdat 2856930 /* The 600-cell */ 2816058 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} rcrq1kus9wr4z0hc145h5qar4zekb3l 2816059 2816058 2026-06-17T05:13:48Z Dc.samizdat 2856930 /* The 600-cell */ 2816059 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_5=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its vertices, over <math>r_{5}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_5</math> chords. The <math>r_5</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_5</math> chords are edges of different 24-cells. The rotational curve over each <math>r_5</math> chord makes five 12° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_5</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} p48fksz22xapj9vzwxvtm7f406lvryy 2816063 2816059 2026-06-17T05:36:43Z Dc.samizdat 2856930 /* The 600-cell */ 2816063 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_{13}=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its <math>r_{4}</math> edges, over <math>r_{13}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_{13}</math> chords. The <math>r_{13}</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_{13}</math> chords are edges of different 24-cells. The rotational curve over each <math>r_{13}</math> chord makes two 30° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_{13}</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} d5vuqfa3h649fcr6u9kzu46gj99w6jz 2816068 2816063 2026-06-17T05:40:37Z Dc.samizdat 2856930 /* The 600-cell */ 2816068 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_{13}=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its <math>r_{4}</math> edges, over <math>r_{13}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_{13}</math> chords. The <math>r_{13}</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_{13}</math> chords are edges of different 24-cells. The rotational curve over each <math>r_{13}</math> chord makes two 30° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_{13}</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} r6a9mdprbqmii8oflbzgmooit73e8mz 2816069 2816068 2026-06-17T05:43:08Z Dc.samizdat 2856930 /* The 600-cell */ 2816069 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 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 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 5² 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 set of 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == 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 [[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 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</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 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the 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, at the vertices of 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, at the vertices of 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. 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 skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} 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 is a construct of 3 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-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes 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 <math>\sqrt{2}</math> edges except opposite pairs. 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}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[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 [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic 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 great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, 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 [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. 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 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 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 two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. 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 between two planes in 4-space. Pairs of square central 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. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, 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.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. 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 the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation 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-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular star figure 2(12,5).svg|100px]]<br>{24/10}=2{12/5} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies the discrete isoclinic rotation of the 600-cell over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon, in invariant central planes containing at least one vertex of the short chord {30}-gon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{11}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{11}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{11}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{11}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_{13}=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its <math>r_{4}</math> edges, over <math>r_{13}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_{13}</math> chords. The <math>r_{13}</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_{13}</math> chords are edges of different 24-cells. The rotational curve over each <math>r_{13}</math> chord makes two 30° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_{13}</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. 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. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally 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 the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 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}} 8sf3gzop0564a6rxndfam9h7fdjqil7 1 326949 2815927 2815856 2026-06-16T13:02:06Z Regliste 3029369 answer to review 2 2815927 wikitext text/x-wiki == Slight modifications of the article == Hello,<br> I imported this page from the Wikipedia article, which I revamped. But since the import, some contributors made helpful comments and edits. I tried to update them all here, but now I stopped and I will just re-import the Wikipedia article when the peer-review process will start. Please notify me when it happens, or re-import it yourself {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 09:48, 13 January 2026 (UTC) ==Peer review 1== {{review |reviewer =Sanjay Ramassamy |Q =Q102641962 |affiliation=Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique |link =https://www.normalesup.org/~ramassamy/index.html.en |date = 1 June 2026 |text = This review article is very well-written, mathematically sound and accessible to people outside the field. I only have minor comments below, most of them typos. I recommend publishing the article once the comments are taken into account. General comment: There are several figures next to the text, but the figures don't seem to be cited in the text. I don't know if this is a journal policy, but it looks a bit unusual to me. Second sentence of the abstract: there is twice ""a new polygon"". Maybe you could rephrase it in a way to eliminate one of the occurrences. E.g. something like ""It defines a new polygon whose vertices are obtained as the intersection points of the shortest diagonals of the initial polygon."" End of first paragraph of the abstract: maybe you could already reference Schwartz's original paper here. Euclidean plane: please capitalize the first letter of ""Euclidean"" throughout the article Section ""On polygons"": ""Finally, it is possible that two diagonals are parallel and not intersect"" -> ""and don't intersect"" Section ""On the moduli space of polygons"": it is the first time that I see the term ""projectivity"". I checked that it was indeed correct, but in all the talks/articles that I have seen on the topic, people rather used ""up to projective transformations"". Section ""Historical elements"", last sentence: it is not too clear what that sentence means. The pentagram map pertains to the field of incidence geometry, like these 3 theorems. What are the further similarities ? Further down in the article, in the section ""Pentagons and hexagons"", there is a similar sentence: ""The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others"". Is it just the case of pentagons and hexagons that resembles these theorems ? Section ""Definition of the map"", first paragraph: it looks strange to cite Weinreich's paper to justify the rather obvious fact that the dimension of the space of n-gons is 2n. More generally, for review articles in WikiJournal, what is the purpose of citations ? Providing a source where something is nicely explained ? Or providing the first source to show some result ? In this article, it seems to be rather the former. Section ""Definition of the map"", second paragraph: ""Taking the intersection of the two..."" -> ""Taking the intersection of two..."" Section ""Twisted polygons"": ""space of twisted n-gon"" -> ""space of twisted n-gons"" ""the dynamic"" -> ""the dynamics"" It comes with a final s even though it is singular, e.g. ""the dynamics is integrable"" Section ""Pentagons and hexagons"": ""The two following facts"" -> ""The following two facts"" Section ""Poncelet polygons"": circumbscribed -> circumscribed Section ""Poncelet polygons"": ""For a convex Poncelet n-gons"" -> n-gon Section ""ab-coordinates"": I would write ""vertices v_k"" and ""vectors V_k"" rather than ""vertices v_k's"" and ""vectors V_k's"" Section ""As a birational map"": you have twice in a row the word pentagram in the first line Section ""The scaling symmetry"": ""an s"" -> ""and s"". Section ""The scaling symmetry"": ""An homogeneous"" -> ""A homogeneous"". Why do you define the notion of weight in this section ? It looks weird because you don't use it immediately, but only towards the end of the next section. It would suggest moving it much closer to the place where you first use it. Section ""The spectral curve"", last sentence: here you write ""algebraic integrability"". In the next sentence it is called ""algebro-geometric integrability"". I prefer the latter formulation. Section ""The spectral curve"": ""some renormalization it"" -> missing ""of"" Section ""Algebro-geometric integrability"": ""in term of"" -> terms Section ""Dimension of the invariant manifold"": ""For a twisted n-gons"" -> ""For twisted n-gons"" Section ""Dimension of the invariant manifold"": what does it mean that the dimension of the invariant tori drops by 3 for closed n-gons ? That it is always n-3 regardless of the parity of n ? Shouldn't invariant tori always be even-dimensional ? Maybe make a separate sentence discussing the closed n-gons case. Section ""Cluster algebras"": rather than ""special cases of cluster algebra"", I would suggest something like ""special cases of discrete dynamical systems powered by cluster algebras"". Because the pentagram map itself is not a cluster algebra. Also, the mutations of the underlying cluster algebra induced by the pentagram map are only a subset of all possible mutations. Section ""Generalizations"": ""description ... as cluster algebras"" -> maybe ""in terms of cluster algebras"" ? Section ""Generalized pentagram maps"": it could be helpful to write that one recovers the original pentagram map by taking d=2, I={2}, J={1}. What surprises me is that for this original pentagram map the set I and J are not equal and yet it is integrable. How is that compatible with the statement that ""the general case is not integrable"" ? Also, just below, the dented pentagram maps provide another class of integrable examples where I and J are not equal. How do you quantify that most cases are not integrable. Section ""Corrugated polygons"": ""they can retrieved"" -> ""they can be retrieved"" ""Grassmannians polygons"" -> ""Grassmannian polygons"" ""the space of Grassmannians Gr(m,md)"" -> ""the Grassmannian space Gr(m,md)"" ""A point in v"" -> ""A point v"" ""general linear group Gl_{md}"" -> ""general linear group GL_{md}"" ""faithfull"" -> faithful ""generically define"" -> ""generically defines"" ""a new point of v"" -> ""a new point v"" }} {{response|1 =Hello, and thanks a lot for the thorough review. I am a bit embarrassed by the numerous typos, they are now fixed. I also reformulated many items following your suggestions. There remains two points I need to answer to. * Indeed, the citation of papers (even for obvious facts) is more frequent than in classical papers. This is because Wikipedia aims to have every statement linked to a reference (see [[w:Wikipedia:Verifiability]]). Some editors take this very seriously (see [https://en.wikipedia.org/wiki/Wikipedia%20talk:WikiProject%20Mathematics/Archive/2025/Dec this discussion]), so I added citations to almost every paragraphs. I guess it could be mitigated for publication. * I clarified the statement about the dimension of invariant manifolds for closed polygons, with one more citation. According to it, they will always be odd-dimensional. Thanks again, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 15:44, 2 June 2026 (UTC)}} == Peer review 2 == {{review |reviewer =Paul Melotti |Q = Q103240269 |affiliation=Université Paris-Saclay |link =https://www.imo.universite-paris-saclay.fr/~paul.melotti/ |date = 11 June 2026 |text = This is a very well-written summary of results on the pentagram map, a fascinating topic that deserved a good presentation in the wikipedia universe. The paper is presented in a clear and coherent way, and I believe it is accessible to non-specialists, provided some minimal background in projective geometry. As far as I could check, the claims are supported by the plentiful references, and they give a good overview of the topic, its history, connections to various topics in mathematics, and modern perspectives. As a general remark, I think the special property of the map T on the spaces of pentagons and hexagons, stated in Section "Periodic orbits on the moduli space", could be stated earlier in the paper, possibly in an informal way. They are quite striking and, in my opinion, motivate the study of the generic transformation. Here are a few minor remarks: - several references to pictures use the phrase "on Figure...", I believe "in Figure..." is more common. - "its interpretation as a cluster algebra" -> maybe "in terms of a cluster algebra", or something similar, would be more precise. - On reference [2] by Gekhtman and Izosimov, "Integrable Systems and Cluster Algebras", the link to sciencedirect in "Works cited" doesn't seem to work when I click it. This might be on my side, but please check the URL. - "for generic polygons on the real projective plane" -> "in" the projective plane seems more common? - "by taking lines and intersections of them" sound a bit weird to me (but I'm not a native speaker so maybe it's okay) - maybe at the beginning of Section "Coordinates for the moduli space", announce that these will allow for nice expressions of the map T in those coordinates (as it is done in the following section). - "This generically makes a quasiperiodic motion." -> "makes" sounds a bit vague to me, maybe "induces a quasiperiodic motion on the corresponding torus" or something. - In the subsection "Grassmannian polygon", second paragraph, I am a bit confused with notations and conventions. If we represent the vector space $v$ by a basis, and put the vectors in columns, we get a matrix of size $md \times m$ and not $m \times md$ right? And then, the action of $GL_{md}$ you are mentioning is simply multiplication on the left? }} [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:30, 15 June 2026 (UTC) {{response|1 = Hello and many thanks for the review. I implemented the changes following your remarks. There was indeed a confusion in the "Grassmannian polygon" section, which is now fixed. Thank you for your vigilance. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 13:01, 16 June 2026 (UTC)}} == Peer review 3 == {{review |reviewer =Richard Evan Schwartz |Q =Q3893370 |affiliation=Brown University |link = |date = 15 June 2026 |text = This article is an update of the wikipedia page for the pentagram map, which I largely wrote myself. (I wrote almost the entire thing because what had been there initially was not very good.) I think that JB did an excellent job updating the pentagram map page. The article hits the main points : classical geometric results, Arnold-Liouville integrability, algebro-geometric integrability, Lax Pairs, connections to cluster algebras, Glick's result about the collapse point, and various generalizations. }} [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:37, 15 June 2026 (UTC) c163r7gkqr31s2njpt938ng46p3066m 2815950 2815927 2026-06-16T14:17:06Z Regliste 3029369 2815950 wikitext text/x-wiki == Slight modifications of the article == Hello,<br> I imported this page from the Wikipedia article, which I revamped. But since the import, some contributors made helpful comments and edits. I tried to update them all here, but now I stopped and I will just re-import the Wikipedia article when the peer-review process will start. Please notify me when it happens, or re-import it yourself {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 09:48, 13 January 2026 (UTC) ==Peer review 1== {{review |reviewer =Sanjay Ramassamy |Q =Q102641962 |affiliation=Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique |link =https://www.normalesup.org/~ramassamy/index.html.en |date = 1 June 2026 |text = This review article is very well-written, mathematically sound and accessible to people outside the field. I only have minor comments below, most of them typos. I recommend publishing the article once the comments are taken into account. General comment: There are several figures next to the text, but the figures don't seem to be cited in the text. I don't know if this is a journal policy, but it looks a bit unusual to me. Second sentence of the abstract: there is twice ""a new polygon"". Maybe you could rephrase it in a way to eliminate one of the occurrences. E.g. something like ""It defines a new polygon whose vertices are obtained as the intersection points of the shortest diagonals of the initial polygon."" End of first paragraph of the abstract: maybe you could already reference Schwartz's original paper here. Euclidean plane: please capitalize the first letter of ""Euclidean"" throughout the article Section ""On polygons"": ""Finally, it is possible that two diagonals are parallel and not intersect"" -> ""and don't intersect"" Section ""On the moduli space of polygons"": it is the first time that I see the term ""projectivity"". I checked that it was indeed correct, but in all the talks/articles that I have seen on the topic, people rather used ""up to projective transformations"". Section ""Historical elements"", last sentence: it is not too clear what that sentence means. The pentagram map pertains to the field of incidence geometry, like these 3 theorems. What are the further similarities ? Further down in the article, in the section ""Pentagons and hexagons"", there is a similar sentence: ""The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others"". Is it just the case of pentagons and hexagons that resembles these theorems ? Section ""Definition of the map"", first paragraph: it looks strange to cite Weinreich's paper to justify the rather obvious fact that the dimension of the space of n-gons is 2n. More generally, for review articles in WikiJournal, what is the purpose of citations ? Providing a source where something is nicely explained ? Or providing the first source to show some result ? In this article, it seems to be rather the former. Section ""Definition of the map"", second paragraph: ""Taking the intersection of the two..."" -> ""Taking the intersection of two..."" Section ""Twisted polygons"": ""space of twisted n-gon"" -> ""space of twisted n-gons"" ""the dynamic"" -> ""the dynamics"" It comes with a final s even though it is singular, e.g. ""the dynamics is integrable"" Section ""Pentagons and hexagons"": ""The two following facts"" -> ""The following two facts"" Section ""Poncelet polygons"": circumbscribed -> circumscribed Section ""Poncelet polygons"": ""For a convex Poncelet n-gons"" -> n-gon Section ""ab-coordinates"": I would write ""vertices v_k"" and ""vectors V_k"" rather than ""vertices v_k's"" and ""vectors V_k's"" Section ""As a birational map"": you have twice in a row the word pentagram in the first line Section ""The scaling symmetry"": ""an s"" -> ""and s"". Section ""The scaling symmetry"": ""An homogeneous"" -> ""A homogeneous"". Why do you define the notion of weight in this section ? It looks weird because you don't use it immediately, but only towards the end of the next section. It would suggest moving it much closer to the place where you first use it. Section ""The spectral curve"", last sentence: here you write ""algebraic integrability"". In the next sentence it is called ""algebro-geometric integrability"". I prefer the latter formulation. Section ""The spectral curve"": ""some renormalization it"" -> missing ""of"" Section ""Algebro-geometric integrability"": ""in term of"" -> terms Section ""Dimension of the invariant manifold"": ""For a twisted n-gons"" -> ""For twisted n-gons"" Section ""Dimension of the invariant manifold"": what does it mean that the dimension of the invariant tori drops by 3 for closed n-gons ? That it is always n-3 regardless of the parity of n ? Shouldn't invariant tori always be even-dimensional ? Maybe make a separate sentence discussing the closed n-gons case. Section ""Cluster algebras"": rather than ""special cases of cluster algebra"", I would suggest something like ""special cases of discrete dynamical systems powered by cluster algebras"". Because the pentagram map itself is not a cluster algebra. Also, the mutations of the underlying cluster algebra induced by the pentagram map are only a subset of all possible mutations. Section ""Generalizations"": ""description ... as cluster algebras"" -> maybe ""in terms of cluster algebras"" ? Section ""Generalized pentagram maps"": it could be helpful to write that one recovers the original pentagram map by taking d=2, I={2}, J={1}. What surprises me is that for this original pentagram map the set I and J are not equal and yet it is integrable. How is that compatible with the statement that ""the general case is not integrable"" ? Also, just below, the dented pentagram maps provide another class of integrable examples where I and J are not equal. How do you quantify that most cases are not integrable. Section ""Corrugated polygons"": ""they can retrieved"" -> ""they can be retrieved"" ""Grassmannians polygons"" -> ""Grassmannian polygons"" ""the space of Grassmannians Gr(m,md)"" -> ""the Grassmannian space Gr(m,md)"" ""A point in v"" -> ""A point v"" ""general linear group Gl_{md}"" -> ""general linear group GL_{md}"" ""faithfull"" -> faithful ""generically define"" -> ""generically defines"" ""a new point of v"" -> ""a new point v"" }} {{response|1 =Hello, and thanks a lot for the thorough review. I am a bit embarrassed by the numerous typos, they are now fixed. I also reformulated many items following your suggestions. There remains two points I need to answer to. * Indeed, the citation of papers (even for obvious facts) is more frequent than in classical papers. This is because Wikipedia aims to have every statement linked to a reference (see [[w:Wikipedia:Verifiability]]). Some editors take this very seriously (see [https://en.wikipedia.org/wiki/Wikipedia%20talk:WikiProject%20Mathematics/Archive/2025/Dec this discussion]), so I added citations to almost every paragraphs. I guess it could be mitigated for publication. * I clarified the statement about the dimension of invariant manifolds for closed polygons, with one more citation. According to it, they will always be odd-dimensional. Thanks again, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 15:44, 2 June 2026 (UTC)}} == Peer review 2 == {{review |reviewer =Paul Melotti |Q = Q103240269 |affiliation=Université Paris-Saclay |link =https://www.imo.universite-paris-saclay.fr/~paul.melotti/ |date = 11 June 2026 |text = This is a very well-written summary of results on the pentagram map, a fascinating topic that deserved a good presentation in the wikipedia universe. The paper is presented in a clear and coherent way, and I believe it is accessible to non-specialists, provided some minimal background in projective geometry. As far as I could check, the claims are supported by the plentiful references, and they give a good overview of the topic, its history, connections to various topics in mathematics, and modern perspectives. As a general remark, I think the special property of the map T on the spaces of pentagons and hexagons, stated in Section "Periodic orbits on the moduli space", could be stated earlier in the paper, possibly in an informal way. They are quite striking and, in my opinion, motivate the study of the generic transformation. Here are a few minor remarks: - several references to pictures use the phrase "on Figure...", I believe "in Figure..." is more common. - "its interpretation as a cluster algebra" -> maybe "in terms of a cluster algebra", or something similar, would be more precise. - On reference [2] by Gekhtman and Izosimov, "Integrable Systems and Cluster Algebras", the link to sciencedirect in "Works cited" doesn't seem to work when I click it. This might be on my side, but please check the URL. - "for generic polygons on the real projective plane" -> "in" the projective plane seems more common? - "by taking lines and intersections of them" sound a bit weird to me (but I'm not a native speaker so maybe it's okay) - maybe at the beginning of Section "Coordinates for the moduli space", announce that these will allow for nice expressions of the map T in those coordinates (as it is done in the following section). - "This generically makes a quasiperiodic motion." -> "makes" sounds a bit vague to me, maybe "induces a quasiperiodic motion on the corresponding torus" or something. - In the subsection "Grassmannian polygon", second paragraph, I am a bit confused with notations and conventions. If we represent the vector space $v$ by a basis, and put the vectors in columns, we get a matrix of size $md \times m$ and not $m \times md$ right? And then, the action of $GL_{md}$ you are mentioning is simply multiplication on the left? }} [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 02:30, 15 June 2026 (UTC) {{response|1 = Hello and many thanks for the review. I implemented the changes following your remarks. There was indeed a confusion in the "Grassmannian polygon" section, which is now fixed. Thank you for your vigilance. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 13:01, 16 June 2026 (UTC)}} == Peer review 3 == {{review |reviewer =Richard Evan Schwartz |Q =Q3893370 |affiliation=Brown University |link = |date = 15 June 2026 |text = This article is an update of the wikipedia page for the pentagram map, which I largely wrote myself. (I wrote almost the entire thing because what had been there initially was not very good.) I think that JB did an excellent job updating the pentagram map page. The article hits the main points : classical geometric results, Arnold-Liouville integrability, algebro-geometric integrability, Lax Pairs, connections to cluster algebras, Glick's result about the collapse point, and various generalizations. }} [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 20:37, 15 June 2026 (UTC) {{response|1 = Hello and thank you for the review. Indeed, you made a massive contribution to the Wikipedia article, as it can be seen by comparing this two versions: [https://en.wikipedia.org/w/index.php?title=Pentagram%20map&oldid=436156794 before] and [https://en.wikipedia.org/w/index.php?title=Pentagram%20map&oldid=438579263 after]. Of course, as stated in the [[WikiJournal User Group/Publishing|guidelines of the journal]], [https://xtools.wmcloud.org/articleinfo/en.wikipedia.org/Pentagram%20map Wikipedia contributors] are also credited. My contribution to reshape it to the standards of the Wikijournal of Science can be seen [https://en.wikipedia.org/w/index.php?title=Pentagram_map&diff=cur&oldid=1317663617 here]. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:17, 16 June 2026 (UTC)}} ruw146v97wcewf42wmr2o9o2wnv337a 2 327407 2816030 2804499 2026-06-16T23:35:45Z Atcovi 276019 /* Future Outlook */ expand 2816030 wikitext text/x-wiki ''<small>Will be moved to [[WikiJournal Preprints/Mental health in Sri Lanka]] - I think the page is too long, so it's making editing it burdensome.</small>'' == Future Outlook == Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A mental health legislation, drafted in 2007, includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be accepted. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated with a lack of an up-to-date legal framework, impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. === Criticism of the Mental Disease Ordinance of 1956 === <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) ede9hfoy2tnhe408pno1dx2z4eifsf2 2816031 2816030 2026-06-16T23:40:04Z Atcovi 276019 Atcovi moved page [[User:Atcovi/sandbox]] to [[User:Atcovi/WikiJournal Preprints/Mental health in Sri Lanka/Future Outlook]] without leaving a redirect: meaningful title page 2816030 wikitext text/x-wiki ''<small>Will be moved to [[WikiJournal Preprints/Mental health in Sri Lanka]] - I think the page is too long, so it's making editing it burdensome.</small>'' == Future Outlook == Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A mental health legislation, drafted in 2007, includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be accepted. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated with a lack of an up-to-date legal framework, impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. === Criticism of the Mental Disease Ordinance of 1956 === <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) ede9hfoy2tnhe408pno1dx2z4eifsf2 2816047 2816031 2026-06-16T23:45:12Z Atcovi 276019 cat(s) 2816047 wikitext text/x-wiki ''<small>Will be moved to [[WikiJournal Preprints/Mental health in Sri Lanka]] - I think the page is too long, so it's making editing it burdensome.</small>'' == Future Outlook == Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A mental health legislation, drafted in 2007, includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be accepted. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated with a lack of an up-to-date legal framework, impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. === Criticism of the Mental Disease Ordinance of 1956 === <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) [[Category:Atcovi's Work]] q9narj3nsg6dlifv8peag6dxny1c6po 0 329829 2816016 2815858 2026-06-16T20:07:51Z Scogdill 1331941 /* Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball */ 2816016 wikitext text/x-wiki = The Irish Aristocracy at the End of the 19th Century = == The Irish Peerage == Minus the people who attended the ball, which are in [[Social Victorians/Irish Aristocracy#Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball|this section, below]]. === Dukes and Duchesses === ==== Duke of Leinster ==== Irish peerage * Gerald FitzGerald, 5th Duke of Leinster (16 August 1851 – 1 December 1893)<ref>{{Cite web|url=https://www.thepeerage.com/p1207.htm#i12063|title=Gerald FitzGerald, 5th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Maurice FitzGerald, 6th Duke of Leinster, 6 years old when he succeeded to the dukedom<ref>{{Cite web|url=https://www.thepeerage.com/p2767.htm#i27667|title=Maurice FitzGerald, 6th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Subsidiary Titles # Marquess of Kildare (Irish peerage), did not attend the ball. # Earl of Kildare (Irish peerage), did not attend the ball. # Earl of Offaly (Irish peerage) # Viscount Leinster of Taplow (GB peerage) # Baron Offaly (Irish peerage) # Baron Kildare of Kildare (UK peerage) === Marquesses and Marchionesses === ==== Marquess Conyngham<ref>{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&oldid=1332742873|journal=Wikipedia|language=en}}</ref> ==== * Did not attend the ball but did attend a number of social events about this time. * Pronounced "''Cunn''ingum."<ref>{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&oldid=1332742873|journal=Wikipedia|language=en}}</ref> * Henry Francis Conyngham, 4th Marquess Conyngham (1857–1897)<ref>"Henry Francis Conyngham, 4th Marquess Conyngham." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71982.</ref> * Victor George Henry Francis Conyngham, 5th Marquess Conyngham (1883–1918)<ref>"Victor George Henry Francis Conyngham, 5th Marquess Conyngham." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71983.</ref> * Subsidiary Titles ** Earl of Conyngham ** Viscount Conyngham ** Viscount Mount Charles ==== Marquess of Donegall ==== * Did not attend the ball. * Subsidiary Titles ** Earl of Donegall, did not attend the ball. ** Viscount Chichester — did not attend the ball; some Chichesters attended social events at about this time. ==== Marquess and Marchioness of Downshire ==== * Arthur Wills John Wellington Trumbull Blundell Hill, 6th Marquess of Downshire (2 July 1871 – 29 May 1918) in 1893 married Katherine Mary ("Kitty") Hare (1872–1959)<ref>{{Cite journal|date=2025-02-10|title=Arthur Hill, 6th Marquess of Downshire|url=https://en.wikipedia.org/w/index.php?title=Arthur_Hill,_6th_Marquess_of_Downshire&oldid=1274976272|journal=Wikipedia|language=en}}</ref> * Did not attend the ball. * Subsidiary Titles ** Earl of Hillsborough, did not attend the ball, also not at any social events described so far. ** Viscount Kilwarlin — 6th, Arthur Wills John Wellington Trumbull Hill (31 March 1874 – 29 May 1918)<ref>"Arthur Wills John Wellington Trumbull '''Hill''', 6th Marquess of Downshire." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page #3810 https://www.thepeerage.com/p3811.htm#i38104.</ref> ==== Marquess of Ely ==== * Did not attend the ball, but members of the Loftus family attended a number of social events at about this time. * 4th Marquess: John Henry Wellington Graham Loftus (15 July 1857 – 3 April 1889)<ref>"John Henry Wellington Graham Loftus, 4th Marquess of Ely." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8545 https://www.thepeerage.com/p8545.htm#i85450.</ref> * 5th Marquess: John Henry Loftus (3 April 1889 – 18 December 1925)<ref>"John Henry Loftus, 5th Marquess of Ely." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8546 https://www.thepeerage.com/p8546.htm#i85459.</ref> * Subsidiary Titles ** Earl of Ely — did not attend the ball. ** Viscount Loftus ==== [[Social Victorians/People/Bective|Marquess and Marchioness of Headfort]] ==== * Did not attend the ball, but a number of people in this family attended many social events at about this time. * Subsidiary Titles ** [[Social Victorians/People/Bective|Earl of Bective]] ** Viscount Headfort<ref name=":1" /> *** 4th: Thomas Taylour (6 December 1870 – 22 July 1894) *** 5th: Geoffrey Thomas Taylour (22 July 1894 – 29 January 1943) *Papers ==== Marquess of Sligo ==== * Did not attend the ball, but many people with the surname Browne attended a number of social events at about this time. * Subsidiary Titles ** Earl of Altamont. Did not attend the ball; did not attend any social events analyzed so far. ** Earl of Clanricarde — Did not attend the ball but did attend a few social events about this time. ** Viscount of Westport<ref name=":1">"Index to Viscounts and Viscountesses." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. https://www.thepeerage.com/index_viscount.htm.</ref> *** 5th: George John Browne (26 January 1845 – 30 December 1896), 5th Marquess *** 6th: John Thomas Browne (30 December 1896 – 30 December 1903), 6th Marquess ==== Marquess of Waterford ==== * John Henry de La Poer Beresford, 5th Marquess of Waterford (1844–1895) * Henry de La Poer Beresford, 6th Marquess of Waterford (1875–1911)<ref>{{Cite journal|date=2026-02-10|title=Henry Beresford, 6th Marquess of Waterford|url=https://en.wikipedia.org/w/index.php?title=Henry_Beresford,_6th_Marquess_of_Waterford&oldid=1337565707|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of the Beresford family were prominent socially at about this time. * Subsidiary Titles ** Viscount Tyrone === Earls and Countesses === ==== Earl of Annesley ==== * Did not attend the ball but did attend a number of social events in the 1890s. * Subsidiary Title ** Viscount Glerawly<ref name=":1" />: 6th: Hugh Annesley (10 August 1874 – 15 December 1908), 6th Earl of Annesley ==== Earl of Bessborough ==== * Frederick George Brabazon Ponsonby, 6th Earl of Bessborough (1815–1895) * Walter William Brabazon Ponsonby, 7th Earl of Bessborough (1821–1906), would have been Viscount Duncannon 1880–1895 * Edward Ponsonby, 8th Earl of Bessborough (1851–1920), would have been Viscount Duncannon 1895–1906 * Did not attend the ball, but the [[Social Victorians/People/Ponsonby|Ponsonby]] family attended many social events at about this time, including mention of Lady Duncannon's school that taught fabric arts. * Subsidiary Titles ** Viscount Duncannon ==== Earl of Caledon ==== * Did not attend the ball but did attend a number of social events about this time. * James Alexander, 4th Earl of Caledon (1846–1898)<ref>{{Cite journal|date=2025-11-21|title=James Alexander, 4th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=James_Alexander,_4th_Earl_of_Caledon&oldid=1323312651|journal=Wikipedia|language=en}}</ref> * Eric James Desmond Alexander, 5th Earl of Caledon (1885–1968), succeeded as earl in 1898.<ref>{{Cite journal|date=2025-11-21|title=Eric Alexander, 5th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=Eric_Alexander,_5th_Earl_of_Caledon&oldid=1323313583|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Caledon ==== Earl of Carrick ==== * Did not attend the ball. ==== Earl Castle Stewart ==== * Did not attend the ball. * 5th Earl: Henry James Stuart-Richardson (12 September 1874 – 5 June 1914)<ref>"Henry James Stuart-Richardson, 5th Earl Castle Stewart." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 2412 https://www.thepeerage.com/p12413.htm#i124125.</ref> * Subsidiary Title ** Viscount Castle Stewart ==== Earl of Cavan ==== * Did not attend the ball. ==== Earl of Clancarty ==== * Did not attend the ball and attended few social events researched so far. * Richard Somerset Le Poer Trench, 4th Earl of Clancarty (1834–1891)<ref>{{Cite journal|date=2026-01-10|title=Richard Trench, 4th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=Richard_Trench,_4th_Earl_of_Clancarty&oldid=1332219771|journal=Wikipedia|language=en}}</ref> * William Frederick Le Poer Trench, 5th Earl of Clancarty (1868–1929)<ref>{{Cite journal|date=2025-11-05|title=William Trench, 5th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=William_Trench,_5th_Earl_of_Clancarty&oldid=1320532351|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Dunlo ==== [[Social Victorians/People/Clanwilliam|Earl and Countess of Clanwilliam]] ==== * Did not attend the ball. * Subsidiary Title ** Viscount Clanwilliam<ref name=":1" />: 4th: Richard James Meade (7 October 1879 – 4 August 1907), 4th Earl ==== Earl of Cork, Earl of Orrery ==== * Cork and Orrery, did attend the ball. ==== Earl of Courtown ==== * Did not attend the ball. ==== Earl of Darnley ==== * John Bligh, 6th Earl of Darnley (1827–1896), British<ref>{{Cite journal|date=2026-02-07|title=John Bligh, 6th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=John_Bligh,_6th_Earl_of_Darnley&oldid=1337113925|journal=Wikipedia|language=en}}</ref> * Edward Bligh, 7th Earl of Darnley (1851–1900), Lord Clifton much of his adult life, "English"<ref>{{Cite journal|date=2026-05-05|title=Edward Bligh, 7th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=Edward_Bligh,_7th_Earl_of_Darnley&oldid=1352607379|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but the Bligh family attended some social events from about this time. * Subsidiary Titles: ** Viscount Darnley ==== Earl of Desmond ==== * Did not attend the ball. ==== [[Social Victorians/People/Donoughmore|Earl of Donoughmore]] ==== * Did not attend the ball but did attend a number of social events about this time. * John Luke George Hely-Hutchinson, 5th Earl of Donoughmore (1848–1900)<ref>{{Cite journal|date=2025-05-01|title=John Hely-Hutchinson, 5th Earl of Donoughmore|url=https://en.wikipedia.org/w/index.php?title=John_Hely-Hutchinson,_5th_Earl_of_Donoughmore&oldid=1288332715|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Donoughmore ==== Earl of Drogheda ==== * Did not attend the ball. * Subsidiary Titles ** Viscount Moore — no evidence of the Viscount or Viscountess Moore at social events at about this time. ==== Earl of Granard ==== * Did not attend the ball. * Bernard Arthur William Patrick Hastings Forbes, 8th Earl of Granard (17 September 1874 – 10 September 1948)[https://en.wikipedia.org/wiki/Bernard_Forbes,_8th_Earl_of_Granard] * Anglo-Irish * Subsidiary Titles ** Bernard Arthur William Patrick Hastings Forbes, styled Viscount Forbes from 1874 to 1889 ==== Earl of Kingston ==== * Did not attend the ball. * Subsidiary Title ** Viscount Kingsborough (of Viscount Kingston of Kingborough, co. Sligo)<ref name=":1" /> *** 8th: Henry Newcomen King-Tenison (21 June 1871 – 13 January 1896) *** 9th: Henry Edwyn King-Tenison (13 January 1896 – 11 January 1946) **Viscount Lorton ==== Earl of Lisburne ==== * Did not attend the ball. * Ernest Augustus Malet Vaughan, 5th Earl of Lisburne (1836–1888)<ref>{{Cite journal|date=2025-12-03|title=Ernest Augustus Malet Vaughan, 5th Earl of Lisburne|url=https://en.wikipedia.org/w/index.php?title=Ernest_Augustus_Malet_Vaughan,_5th_Earl_of_Lisburne&oldid=1325511612|journal=Wikipedia|language=en}}</ref> ** Owned a lot of land in Cardiganshire, Wales ** Conservative, but withdrew from politics * George Henry Arthur Vaughan, 6th Earl of Lisburne (1862–1899) * Ernest Edmund Henry Malet Vaughan, 7th Earl of Lisburne (1892–1965) ** Welsh nobleman, of Trawsgoed, Cardiganshire. 7 years old when he succeeded to the earldom ==== Earl of Longford ==== * Did not attend the ball. ==== Earl and Countess of Meath ==== * Did not attend the ball. ==== Earl of Mexborough ==== * Did not attend the ball ==== Earl of Mornington ==== * Subsidiary title of the Duke of Wellington (in the peerage of the UK). ==== Earl of Normanton ==== * Did not attend the ball, but did attend some social events in the 1880s and 1890s. * James Charles Herbert Welbore Ellis Agar, 3rd Earl of Normanton (1818–1896)<ref>{{Cite journal|date=2025-10-06|title=James Agar, 3rd Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=James_Agar,_3rd_Earl_of_Normanton&oldid=1315461436|journal=Wikipedia|language=en}}</ref> * Sidney James Agar, 4th Earl of Normanton (1865–1933)<ref>{{Cite journal|date=2026-05-19|title=Sidney James Agar, 4th Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=Sidney_James_Agar,_4th_Earl_of_Normanton&oldid=1355064165|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Somerton ==== Earl of Portarlington ==== * Did not attend the ball. Members of this family attended a few social events at about this time. * Subsidiary Title ** Viscount Carlow<ref name=":1" /> *** 5th: Lionel Seymour William Dawson-Damer (1 March 1889 – 17 December 1892), Earl of Portarlington *** 6th: Lionel George Henry Seymour Dawson-Damer (17 December 1892 – 31 August 1900) ==== Earl of Roden ==== * Did not attend the ball. * Subsidiary Title ** Viscount Jocelyn<ref name=":1" /> *** 6th: John Strange Jocelyn (9 January 1880 – 3 July 1897) *** 7th: William Henry Jocelyn (3 July 1897 – 23 January 1910) ==== Earl of Shannon ==== * Did not attend the ball. ==== Earl of Shelburne ==== * Subsidiary title of the Marquess of Lansdowne (in the peerage of Great Britain). * Did not attend the ball, and did not attend any social events analyzed so far. ==== Earl of Tyrone ==== * Did not attend ==== Earl of Waterford ==== * Not a subsidiary title of the Marquess of Waterford but of the Earl of Shrewsbury in the peerage of England. ==== Earl of Westmeath ==== * Did not attend the ball. ==== Earl of Winterton ==== * Did not attend the ball. === Viscounts and Viscountesses === ==== Viscount Ashbrook ==== * William Spencer Flower, 7th Viscount Ashbrook (1830–1906)<ref>{{Cite journal|date=2025-12-01|title=Viscount Ashbrook|url=https://en.wikipedia.org/w/index.php?title=Viscount_Ashbrook&oldid=1325071512|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, has no social presence at about this time. ==== Viscount Banger ==== * Did not attend the ball but attended a few social events at about this time. * Edward Ward, 4th Viscount Bangor (1827–1881)<ref>{{Cite journal|date=2026-03-16|title=Edward Ward, 4th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Edward_Ward,_4th_Viscount_Bangor&oldid=1343882576|journal=Wikipedia|language=en}}</ref> * Henry William Crosbie Ward, 5th Viscount Bangor (1828–1911)<ref>{{Cite journal|date=2026-03-02|title=Henry Ward, 5th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Henry_Ward,_5th_Viscount_Bangor&oldid=1341354058|journal=Wikipedia|language=en}}</ref> ==== Viscount Boyne ==== * Did not attend the ball, but did attend a number of events at about this time. ==== Viscount Callan ==== * Did not attend the ball, and does not have much if any social presence at about this time. * The Viscount Callan is a subsidiary title of the Earl of Denbigh in the Peerage of England. ==== Viscount Charlemont ==== * Did not attend the ball. * Colonel James Alfred Caulfeild, 7th Viscount Charlemont (20 March 1830 – 4 July 1913), Irish<ref>{{Cite journal|date=2026-05-02|title=James Caulfeild, 7th Viscount Charlemont|url=https://en.wikipedia.org/w/index.php?title=James_Caulfeild,_7th_Viscount_Charlemont&oldid=1352129469|journal=Wikipedia|language=en}}</ref> * Unionist ==== Viscount Chetwynd ==== * Does not seem to have attended the ball, but Chetwynds were socially very active at about this time. * Godfrey John Boyle Chetwynd, 8th Viscount Chetwynd (1863–1936), British<ref>{{Cite journal|date=2026-05-24|title=Godfrey Chetwynd, 8th Viscount Chetwynd|url=https://en.wikipedia.org/w/index.php?title=Godfrey_Chetwynd,_8th_Viscount_Chetwynd&oldid=1355878192|journal=Wikipedia|language=en}}</ref> ==== Viscount de Vesci ==== * Did not attend the ball but attended several social events at about this time. * 4th Viscount de Vesci: John Robert William Vesey (23 December 1875 – 6 July 1903)<ref name=":1" /> * "The family seat was Abbeyleix House, near Abbeyleix, County Laois."<ref>{{Cite journal|date=2026-02-09|title=Viscount de Vesci|url=https://en.wikipedia.org/w/index.php?title=Viscount_de_Vesci&oldid=1337491855|journal=Wikipedia|language=en}}</ref> ==== Viscount Dillon ==== * Did not attend the ball, but several Dillons attended other social events at about this time. ==== Viscount Doneraile<ref>{{Cite journal|date=2026-01-16|title=Viscount Doneraile|url=https://en.wikipedia.org/w/index.php?title=Viscount_Doneraile&oldid=1333262628|journal=Wikipedia|language=en}}</ref> ==== * Did not attend the ball, but did attend the Warwick Bal Poudré and few other social events at about this time. * Hayes St Leger, 4th Viscount Doneraile (1818–1887) * Richard Arthur St Leger, 5th Viscount Doneraile (1825–1891) * Edward St Leger, 6th Viscount Doneraile (1866–1941) ==== [[Social Victorians/People/Downe|Viscount Downe]] ==== * Did not attend the ball but attended many social events at about this time. * Major-General Hugh Richard Dawnay, 8th Viscount Downe (20 July 1844 – 21 January 1924)<ref>{{Cite journal|date=2026-03-24|title=Hugh Dawnay, 8th Viscount Downe|url=https://en.wikipedia.org/w/index.php?title=Hugh_Dawnay,_8th_Viscount_Downe&oldid=1345146095|journal=Wikipedia|language=en}}</ref> * British Army general ==== Viscount Ferrard ==== * See Viscount Massereene, below. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard. ==== Viscount Fitzmaurice ==== * A subsidiary title of the Marquess of Lansdowne (in the Peerage of Great Britain). * 6th Viscount Fitzmaurice, Henry Charles Keith Petty-FitzMaurice (5 July 1866 – 3 June 1927)<ref>"Henry Charles Keith Petty-FitzMaurice, 5th Marquess of Lansdowne." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 958 https://www.thepeerage.com/p959.htm#i9586.</ref> ==== Viscount Gage ==== * Henry Charles Gage, 5th Viscount Gage (1854–1912)<ref>{{Cite journal|date=2025-06-21|title=Viscount Gage|url=https://en.wikipedia.org/w/index.php?title=Viscount_Gage&oldid=1296646030|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. ==== Viscount Galway ==== * George Edmund Milnes Monckton-Arundell, 7th Viscount Galway (1844–1931), British conservative<ref>{{Cite journal|date=2025-08-08|title=George Monckton-Arundell, 7th Viscount Galway|url=https://en.wikipedia.org/w/index.php?title=George_Monckton-Arundell,_7th_Viscount_Galway&oldid=1304770631|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but Viscount and Viscountess Galway attended many social events at about this time. * Subsidiary Title ** Baron Monckton (in the Peerage of the United Kingdom) ==== Viscount Gormanston ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. ==== [[Social Victorians/People/Gort|Viscount Gort]] ==== * Did not attend the ball, but attended some social events at about this time. * Standish Prendergast Vereker, 4th Viscount Gort (1819–1900)<ref>"Standish Prendergast Vereker, 4th Viscount Gort." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 4626 https://www.thepeerage.com/p4627.htm#i46261.</ref> * John Gage Prendergast Vereker, 5th Viscount Gort (1849–1902)<ref>{{Cite journal|date=2025-05-28|title=John Vereker, 5th Viscount Gort|url=https://en.wikipedia.org/w/index.php?title=John_Vereker,_5th_Viscount_Gort&oldid=1292670203|journal=Wikipedia|language=en}}</ref> ==== Viscount Grandison ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. * The Viscount Grandison is a subsidiary title of the Earl of Jersey in the Peerage of England. ==== Viscount Grimston ==== * Subsidiary title of the Earl of Verulam (in the Peerage of the United Kingdom) * Did not attend the ball, but a number of members of this family attended social events at about this time. ==== Viscount Harberton ==== * Did not attend the ball; Viscountess Harberton is mentioned once in social events at about this time so far. * James Spencer Pomeroy, 6th Viscount Harberton (1836–1912)<ref>"James Spencer Pomeroy, 6th Viscount Harberton." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 4315 https://www.thepeerage.com/p43151.htm#i431502.</ref> * Florence Wallace Pomeroy, Viscountess Harberton (1843–1911), suffragette, cyclist, President of the Rational Dress Society<ref>{{Cite journal|date=2026-03-12|title=Florence Wallace Pomeroy, Viscountess Harberton|url=https://en.wikipedia.org/w/index.php?title=Florence_Wallace_Pomeroy,_Viscountess_Harberton&oldid=1343082631|journal=Wikipedia|language=en}}</ref> ==== Viscount Lifford ==== * Did not attend the ball; the only social event at about this time so far is the Queen's Diamond Jubilee garden party. * James Hewitt, 4th Viscount Lifford (1811–1887)<ref>{{Cite journal|date=2025-09-11|title=James Hewitt, 4th Viscount Lifford|url=https://en.wikipedia.org/w/index.php?title=James_Hewitt,_4th_Viscount_Lifford&oldid=1310741456|journal=Wikipedia|language=en}}</ref> * James Wilfrid Hewitt, 5th Viscount Lifford (12 October 1837 – 20 March 1913)<ref>"James Wilfrid Hewitt, 5th Viscount Lifford." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 2227 https://www.thepeerage.com/p22271.htm#i222701.</ref> ==== Earl of Listowel ==== * Pronounced "Lish-''toe''-ell."<ref>{{Cite journal|date=2024-10-15|title=Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Listowel&oldid=1251322273|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but hosted and attended social events at about this time. * William Hare, 3rd Earl of Listowel (1833–1924)<ref>{{Cite journal|date=2026-04-17|title=William Hare, 3rd Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=William_Hare,_3rd_Earl_of_Listowel&oldid=1349570352|journal=Wikipedia|language=en}}</ref>, Irish peer * Subsidiary Title ** Viscount Ennismore and Listowel ** Baron Ennismore ==== Viscount Massereene ==== * Did not attend the ball but did attend a few events at about this time. See Viscount Ferrard, above. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard. * Anglo-Irish * Clotworthy John Eyre Skeffington, 11th Viscount Massereene (9 October 1842 – 26 June 1905)<ref>{{Cite journal|date=2024-11-23|title=Clotworthy Skeffington, 11th Viscount Massereene|url=https://en.wikipedia.org/w/index.php?title=Clotworthy_Skeffington,_11th_Viscount_Massereene&oldid=1259199982|journal=Wikipedia|language=en}}</ref> and 4th Viscount Ferrard (28 April 1863 – 26 June 1905) ==== Viscount Molesworth ==== * Did not attend the ball, but attended the Warwick Bal Poudré and a number of other social events at about this time. * Samuel Molesworth, 8th Viscount Molesworth (1829–1906), may have been a Quaker ==== Viscount Monck ==== * Did not attend the ball, but attended a number of social events at about this time. * Charles Stanley Monck, 4th Viscount Monck (1819–1894)<ref>{{Cite journal|date=2026-04-05|title=Charles Monck, 4th Viscount Monck|url=https://en.wikipedia.org/w/index.php?title=Charles_Monck,_4th_Viscount_Monck&oldid=1347311992|journal=Wikipedia|language=en}}</ref>, British * Henry Power Charles Stanley Monck, 5th Viscount Monck (1849–1927)<ref>"Henry Power Charles Stanley Monck, 5th Viscount Monck of Ballytrammon." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 3880 https://www.thepeerage.com/p3881.htm#i38802.</ref> ==== Viscount Mountgarret ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. ==== [[Social Victorians/People/Powerscourt|Viscount Powerscourt]] ==== * Mervyn Wingfield, 7th Viscount Powerscourt (1836–1904)<ref name=":0">{{Cite journal|date=2026-02-18|title=Mervyn Wingfield, 7th Viscount Powerscourt|url=https://en.wikipedia.org/w/index.php?title=Mervyn_Wingfield,_7th_Viscount_Powerscourt&oldid=1339057453|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. * Subsidiary Title ** Baron Powerscourt (in the Peerage of the United Kingdom), 1885<ref name=":0" /> ==== Viscount Southwell ==== * Did not attend the ball, though the Viscount and Viscountess attended a few social events at about this time. * 5th<ref name=":1" />: Arthur Robert Pyers Southwell (26 April 1878 – 5 October 1944)<ref>"Arthur Robert Pyers Southwell, 5th Viscount Southwell of Castle Mattress." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page https://www.thepeerage.com/p7550.htm#i75497.</ref> ==== Viscount Valentia ==== * Did not attend the ball, attended some social events at about this time. Was on the Welcome Council for the 1887 American Exhibition. === Barons and Baronesses === Not all the barons extant at the end of the 19th century and listed on the Wikipedia [[wikipedia:Peerage_of_Ireland|Peerage of Ireland]] page are here — only the ones who were active socially. ==== Baron Conway and Killultagh ==== * Did not attend the ball, but people from the Conway and Seymour families attended a number of social events at about this time. * Subsidiary title of the Marquess of Hertford (in the Peerage of England and Great Britain). * Francis Hugh George Seymour, 5th Marquess of Hertford (1812–1884)<ref>{{Cite journal|date=2026-04-05|title=Francis Seymour, 5th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Francis_Seymour,_5th_Marquess_of_Hertford&oldid=1347294689|journal=Wikipedia|language=en}}</ref> * Hugh de Grey Seymour, 6th Marquess of Hertford (1843–1912)<ref>{{Cite journal|date=2026-04-05|title=Hugh Seymour, 6th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Hugh_Seymour,_6th_Marquess_of_Hertford&oldid=1347303090|journal=Wikipedia|language=en}}</ref> ==== Baron Digby ==== * Did not attend the ball, but people from this family attended a number of social events at about this time. * Edward St Vincent Digby, 9th and 3rd Baron Digby (1809–1889)<ref>{{Cite journal|date=2025-12-15|title=Edward Digby, 9th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_9th_Baron_Digby&oldid=1327712265|journal=Wikipedia|language=en}}</ref> * Edward Henry Trafalgar Digby, 10th and 4th Baron Digby (1846–1920)<ref>{{Cite journal|date=2026-01-26|title=Edward Digby, 10th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_10th_Baron_Digby&oldid=1334892957|journal=Wikipedia|language=en}}</ref> ==== Baron Inchiquin ==== * Did not attend the ball, but people from this family attended a number of social events at about this time. * Edward Donough O'Brien, 14th Baron Inchiquin (1839–1900)<ref>{{Cite journal|date=2026-04-28|title=Edward O'Brien, 14th Baron Inchiquin|url=https://en.wikipedia.org/w/index.php?title=Edward_O%27Brien,_14th_Baron_Inchiquin&oldid=1351543832|journal=Wikipedia|language=en}}</ref> == Peerage of the United Kingdom of Great Britain and Ireland == After the forced 1801 Act of Union. === Earls and Countesses === ==== Earl of Limerick ==== * Did not attend the ball, but did attend a number of events at about this time. ==== Earl of Norbury ==== * Did not attend the ball, but attended some social events at about this time. * Subsidiary Title ** Baron Norbury ==== Earl of Ranfurly ==== * Did not attend the ball, and they have a small social presence in the newspapers in the 1880s and 1890s. ==== Earl of Rosse ==== * Did not attend the ball, but did attend a few events at about this time. == Peerage of the United Kingdom == * Lurgan == Irish Nationalists == == Irish Unionists == == Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball == ==== [[Social Victorians/People/Abercorn|Duke and Duchess of Abercorn]] ==== * This dukedom is in the peerage of the United Kingdom of Great Britain and Ireland * James Hamilton, 1st Duke of Abercorn (1811–1885), elder son of Lord Hamilton, "styled Viscount Hamilton from 1814 to 1818 and The Marquess of Abercorn from 1818 to 1868, was a Conservative statesman who twice served as Lord Lieutenant of Ireland."<ref>{{Cite journal|date=2026-04-05|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&oldid=1347253763|journal=Wikipedia|language=en}}</ref> * James Hamilton, 2nd Duke of Abercorn (1838–1913), eldest son of the 1st Duke, "styled Viscount Hamilton until 1868 and Marquess of Hamilton from 1868 to 1885, was a British nobleman, courtier, and diplomat."<ref>{{Cite journal|date=2026-01-25|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&oldid=1334676058|journal=Wikipedia|language=en}}</ref> * The Hamilton who became the 3rd duke attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did a few other members of this family. * Subsidiary Titles ** Marquess of Abercorn ** Viscount Hamilton ** Viscount Strabane, county Tyrone *Papers **PRONI for the Abercorn papers [GB 0255 PRONI/D623] **Some individuals' papers (the Tighe Hamilton Howard papers, https://iar.ie/archive/tighe-hamilton-howard-papers) from the Hamilton family are in the National Library of Ireland. "An item level catalogue is available online. These papers form part of the Wicklow Papers (Collection List 69) that are held in the Department of Manuscripts at the National Library of Ireland." ***VII. Sarah Howard Papers, 1830-1887. ******* VII.ii. Letters from Sarah Howard to her husband the Hon. Rev. Francis Howard, [n.d.] Call number: '''MS 38,639/2/2''' ******* VII.iii. Correspondence between Sarah Howard and her daughter Lady Caroline Howard, ca. 1851 - ca. 1891. Call number: '''MS 38,639/2/3''' ****VII.iv. Correspondence between Sarah Howard and her son Charles Howard, 5th Earl of Wicklow, 1853-ca.1870. Call number: '''MS 38,639/2/4''' ****VII.v. Correpondence between Sarah Howard and her son Cecil Howard, 6th Earl of Wicklow, ca. 1855-1876. Call number: '''MS 38,639/2/5''' ******* VII.vi. Correspondence between Sarah Howard and her daughters Lady Louisa and Lady Alice Howard, 1855-ca. 1877. Call number: '''MS 38,639/2/6''' ******* VII.vix. Additional correspondence of Sarah Howard of Wingfield, Bray Co. Wicklow, 1865-1887. Call number: '''MS 38,639/2/9''' ***VIII. Lady Caroline Howard Papers, 1852-1919. ****VIII.i. Correspondence between Lady Caroline Howard and her brother Charles, Earl of Wicklow, 1852-1880. Call number: '''MS 38,639/2/11''' ****VIII.iv. Additional correspondence of Lady Caroline Howard, 1868-1919. Call number: '''MS 38,639/2/14''' ****VIII.v. Additional papers of Lady Caroline Howard, 1900. Call number: '''MS 38,639/2/15''' ***IX. Additional Howard family correspondence, 1773-1900. ****IX.vii. Correspondence and papers of Lady Louisa Howard, 1856-1907. Call number: '''MS 38,639/2/22''' ******* IX.viii. Correspondence and papers of Lady Alice Howard, [n.d.] Call number: '''MS 38,639/2/23''' ***XI. Other papers, 1737-1913. ****XI.i. Miscellaneous correspondence, 1753-1891. Call number: '''MS 38,639/2/27''' ***Wicklow Papers ****** Journals of Lady Caroline Howard including some accounts of her tours abroad, 1873 Jan. - March, 1875 Aug. - Sept., & 1882 Jan. - April. Call number: '''MS 3586-3588''' ****** Diaries of Lady Louisa Howard including accounts of her travels on the Continent, 1862 Oct. - 1869 June, 1871 April - 1873 April and 1877 Oct. - 1883 July. Call number: '''MS 3589-3593''' ****Diaries of Lady Caroline Howard, 1862 Oct. - 1870 May. Call number: '''MS 3594-3599''' ******* Diaries of Lady Alice Howard, Shelton Abbey and Bray, Co. Wicklow, 1874-1922. Call number: '''MS 3600-3625''' ***** Journals of Lady Alice Howard, including account of tours on the Continent, 1860 June - Oct, 1865 Aug. - 1866 Feb., 1869 Nov. - 1870 Nov. Call number: '''MS 4793-4795''' *** ==== [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] ==== * The Marquess and Marchioness attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, she led one of the courts as Maria Thérèse, plus two of their children attended, one of whom is Viscount Castlereagh. * Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry<ref>"Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12772.</ref> * Lady Theresa Susey Helen Chetwynd-Talbot, Marchioness of Londonderry<ref>"Lady Theresa Susey Helen Chetwynd-Talbot." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12771.</ref> * Subsidiary Titles ** [[Social Victorians/People/Londonderry|Earl of Londonderry]] ** Viscount Castlereagh — Charles Stewart Henry Vane-Tempest-Stewart (6 November 1884 – 8 February 1915) *Papers **In PRONI [GB 0255 PRONI/D2846]: "The Theresa, Lady Londonderry Papers comprise c.4,600 papers and 15 volumes of diaries, scrapbooks, etc, 1858-1919, mainly of Theresa, Marchioness of Londonderry (1856-1919), wife/widow of the 6th Marquess, but including some papers of the 6th Marquess himself, of and about his mother, Mary Cornelia, widow of the 5th Marquess, and of his brothers Lords Henry and Herbert Vane-Tempest."<ref>{{Cite web|url=https://iar.ie/archive/theresa-lady-londonderry-papers/|title=Theresa, Lady Londonderry Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> **In PRONI [GB 0255 PRONI/D3099]: the "Papers of the 7th Marquess of Londonderry and his wife Edith" collection also hold the papers of Edith's father, [[Social Victorians/People/Henry Chaplin|Henry, 1st Viscount Chaplin]], who attended the ball, as did she and a brother. [D3099/1 Henry, 1st Viscount Chaplin, father-in-law of 7th Marquess of Londonderry. Political and personal papers; D3099/3 Edith Helen Chaplin, wife of 7th Marquess of Londonderry. Personal letters and papers]<ref>{{Cite web|url=https://iar.ie/archive/papers-7th-marquess-londonderry-wife-edith/|title=Papers of the 7th Marquess of Londonderry and his wife Edith|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> ==== [[Social Victorians/People/Lucan|Earl of Lucan]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, and the family attended a number of social events at this time. * Papers: Irish Archives Resource has one listing for Lucan, but it doesn't seem to be relevant: too late and not about the family. ==== [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] ==== * The marchioness and her daughters attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, though nobody mentions the Marquess. * James Edward Butler, 3rd Marquess of Ormonde and 21st Earl of Ormonde (1844–1919)<ref>{{Cite journal|date=2026-05-03|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&oldid=1352334266|journal=Wikipedia|language=en}}</ref> Now extinct; earldom dormant. Castle Kilkenny was their manor, but they don't appear to have any papers. * Subsidiary Titles * Papers: Irish Archives Resource has one listing, but it's not about the family, the name of a road uses the word ''Ormonde''. ==== [[Social Victorians/People/Antrim|Earl of Antrim]] ==== * The earl and countess did not attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, but two of his brothers did. * Papers ** [https://iar.ie/archive/earl-antrim-estate-papers/ Estate papers of the Earls of Antrim] [GB 0255 PRONI/D2977] are in PRONI. I don't see personal papers listed, but the collection has 50,000 documents 1603–1967. ** Also "D4091 Papers of Sir Schomberg MacDonnell, Louisa Countess of Antrim and the Stuart family of Dalness. MIC615 The diaries of Louisa, Countess of Antrim."<ref>{{Cite web|url=https://iar.ie/archive/earl-antrim-estate-papers/|title=Earl of Antrim Estate Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> ==== [[Social Victorians/People/Arran|Earl of Arran]] ==== * Attended the ball. * Subsidiary Titles ** Viscount Sudley: 5th: Arthur Saunders William Charles Fox Gore (25 Jun 1884-14 Mar 1901), 5th Earl of Arran<ref name=":1" /> *Papers ==== [[Social Victorians/People/Belmore|Earl Belmore]] ==== * Did not attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, although [[Social Victorians/People/Rowton|Montagu Lowry-Corry, 1st Baron Rowton]] did, but did attend a number of social events about this time. * 4th Earl: Somerset Richard Lowry-Corry (17 Dec 1845-6 Apr 1913)<ref>{{Cite journal|date=2026-04-17|title=Somerset Lowry-Corry, 4th Earl Belmore|url=https://en.wikipedia.org/w/index.php?title=Somerset_Lowry-Corry,_4th_Earl_Belmore&oldid=1349375684|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Belmore (though the subsidiary title for the heir apparent is Viscount Corry?) *Papers: Belmore Papers [GB 0255 PRONI/D3007]<ref>{{Cite web|url=https://iar.ie/archive/belmore-papers/|title=Belmore Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-07}}</ref> **D3007/B Rentals and account books (estate, household and personal papers) **D3007/F Curiosa and personal ephemera **D3007/I Private and family letters to Honoria Gladstone, Countess Belmore **D3007/Y Letters and papers of Viscount Corry and the Hon. Cecil Corry, later 5th and 6th Earls Belmore respectively **D3007/Z Family and other photographs ==== [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] ==== * The [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] and Countess of Dunraven, and their daughter Lady Aileen May Wyndham-Quin attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl (1841–1926)<ref>{{Cite journal|date=2026-05-22|title=Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl|url=https://en.wikipedia.org/w/index.php?title=Windham_Wyndham-Quin,_4th_Earl_of_Dunraven_and_Mount-Earl&oldid=1355461019|journal=Wikipedia|language=en}}</ref>, Anglo-Irish * Papers ==== [[Social Victorians/People/Cole|Earl and Countess of Enniskillen]] ==== * The Earl and Countess and a daughter attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. Papers in PRONI. * Subsidiary Title ** 4th Viscount Enniskillen: Lowry Egerton Cole (12 November 1886 – 28 April 1924)<ref name=":1" /> *Papers ==== [[Social Victorians/People/Crichton|Earl of Erne]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * The newspapers were very inconsistent in the spelling of the family name Crichton. * Subsidiary Title ** Viscount Erne<ref name=":1" /> *** 3rd Earl of Erne: John Crichton (10 June 1842 – 3 October 1885) *** 4th Earl of Erne: John Henry Crichton (3 October 1885 – 2 December 1914) *Papers: in PRONI. ==== [[Social Victorians/People/Gosford|Earl of Gosford]] ==== * The Earl and Countess of Gosford attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did a son and a daughter. They attended many social events at about this time. * Subsidiary Title ** Viscount Gosford of Market Hill, co. Armagh<ref name=":1" /> *** 5th Earl of Gosford: Archibald Brabazon Sparrow Acheson (15 June 1864 – 11 April 1922) *Papers ==== Earl of Kerry ==== * Subsidiary title of the [[Social Victorians/People/Lansdowne|Marquess of Lansdowne]] (in the peerage of Great Britain). Attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Subsidiary Titles ** Viscount Clanmaurice *Papers ==== [[Social Victorians/People/Kilmorey|Earl of Kilmorey]] ==== * Anglo-Irish * Nellie Countess of Kilmorey attended the ball; Francis, 3rd Earl was alive at the time, did he attend? Both he and she attended a number of social events from about this time. * Papers ==== [[Social Victorians/People/Mayo|Earl of Mayo]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Viscount Mayo of Monycrower, co. Mayo<ref name=":1" /> ** 7th Earl of Mayo: Dermot Robert Wyndham Bourke (8 February 1872 – 31 December 1927) *Papers ==== [[Social Victorians/People/Midleton|Viscount Midleton]] ==== * Some people from this family seem to have attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House as well as many other social events at about this time. * William Brodrick, 8th Viscount Midleton (6 January 1830 – 18 April 1907), "Irish peer, landowner and Conservative politician in both Houses of Parliament"<ref>{{Cite journal|date=2025-01-05|title=William Brodrick, 8th Viscount Midleton|url=https://en.wikipedia.org/w/index.php?title=William_Brodrick,_8th_Viscount_Midleton&oldid=1267418489|journal=Wikipedia|language=en}}</ref> * Sight and hearing disabilities caused by intermarriage. A daughter became a Republican. * Papers ==== [[Social Victorians/People/Lurgan|Baron Lurgan]] ==== * The Baron, his wife and probably his uncle attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. ** Emily Lady Lurgan ** William Brownlow, Baron Lurgan ** Hon. Cecil Brownlow * Papers, PRONI<ref>{{Cite web|url=https://iar.ie/archive/brownlow-papers/|title=Brownlow Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-07}}</ref> ==== Baron Carrington ==== * [[Social Victorians/People/Carrington|Charles Robert Wynn-Carington, 1st Marquess of Lincolnshire]] (1843–1928) attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Baron Carrington is a subsidiary title of the Marquess of Lincolnshire (created in 1912; Earl Carrington created in 1895).<ref>{{Cite journal|date=2026-05-20|title=Baron Carrington|url=https://en.wikipedia.org/w/index.php?title=Baron_Carrington&oldid=1355207880|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Dufferin and Claneboye<ref>{{Cite journal|date=2026-02-07|title=Baron Dufferin and Claneboye|url=https://en.wikipedia.org/w/index.php?title=Baron_Dufferin_and_Claneboye&oldid=1337113957|journal=Wikipedia|language=en}}</ref> ==== * Members of this family did attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House as well as many social events at about this time. * [[Social Victorians/People/Hamilton Temple Blackwood|Frederick Temple Hamilton-Temple-Blackwood]], 1st Marquess of Dufferin and Ava (1826–1902)<ref>{{Cite journal|date=2026-05-27|title=Frederick Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava|url=https://en.wikipedia.org/w/index.php?title=Frederick_Hamilton-Temple-Blackwood,_1st_Marquess_of_Dufferin_and_Ava&oldid=1356387854|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Garvagh ==== * [[Social Victorians/People/Garvagh|Florence Canning, Lady Garvagh]] attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Charles John Spencer George Canning, 3rd Baron Garvagh (1852–1915)<ref>{{Cite journal|date=2026-02-06|title=Baron Garvagh|url=https://en.wikipedia.org/w/index.php?title=Baron_Garvagh&oldid=1336941309|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Rossmore of Monaghan ==== * A [[Social Victorians/People/Naylor|Miss Naylor]] (Lady Rossmore's sister) of this family attended the ball. * Derrick Warner William Westenra, 5th Baron Rossmore (1853–1921)<ref>{{Cite journal|date=2024-08-27|title=Derrick Westenra, 5th Baron Rossmore|url=https://en.wikipedia.org/w/index.php?title=Derrick_Westenra,_5th_Baron_Rossmore&oldid=1242602083|journal=Wikipedia|language=en}}</ref> * Papers == References == {{reflist}} htp2c7y4lmi2wgkid76j814x52ui2v3 0 330218 2815920 2815915 2026-06-16T12:33:57Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2815920 wikitext text/x-wiki The following families comprise the superfamily Apoidea: *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] {{BookCat}} j862bir5moo1yq1v6d0rc2xfbb73fg6 2815965 2815920 2026-06-16T15:35:19Z Alandmanson 1669821 2815965 wikitext text/x-wiki The following families comprise the superfamily Apoidea: *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] {{BookCat}} khaxi86hhgd0g06l4yrcvcakjty4z99 2815966 2815965 2026-06-16T15:36:20Z Alandmanson 1669821 2815966 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] {{BookCat}} g0i2jsop2dp5kief2q5gqoal77n8f6u 2815999 2815966 2026-06-16T18:53:48Z Alandmanson 1669821 /* Apoidea */ 2815999 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families) }} }} }} }} }} }} }} {{BookCat}} h9rwupffh9736caqb0mn6yvjudeh62f 2816001 2815999 2026-06-16T19:00:38Z Alandmanson 1669821 /* Apoidea */ 2816001 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila). {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families) }} }} }} }} }} }} }} {{BookCat}} dxrv9ap1v21tci8yeu8wnj9afvzsqig 2816004 2816001 2026-06-16T19:09:10Z Alandmanson 1669821 /* Apoidea */ 2816004 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families) }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] {{BookCat}} clxaj5xesj7gjinti2uh2zmywx2k9hf 2816005 2816004 2026-06-16T19:10:21Z Alandmanson 1669821 /* Apoidea */ 2816005 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families) }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} 5bncpf95r2ep62krxvxino6lael1iti 2816006 2816005 2026-06-16T19:12:50Z Alandmanson 1669821 2816006 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} n75bxc7n2o2h5q759apx47mjcyrs9sg 2816007 2816006 2026-06-16T19:13:49Z Alandmanson 1669821 /* Apoidea */ 2816007 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} f59d89y0sq7g1cvj230lh41anpq9yf0 2816008 2816007 2026-06-16T19:17:34Z Alandmanson 1669821 /* Apoidea */ 2816008 wikitext text/x-wiki The following families comprise the superfamily Apoidea: =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} ac4kck0bztzcwwh316jal051h0cu4ot 2816009 2816008 2026-06-16T19:19:05Z Alandmanson 1669821 2816009 wikitext text/x-wiki =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} a24s5u56rgb277ca3z3yrpr7lo3iv4w 2816010 2816009 2026-06-16T19:21:05Z Alandmanson 1669821 /* Apoidea */ 2816010 wikitext text/x-wiki =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=[[Mellinidae]] (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=[[Entomosericidae]] (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} crfzw8pau7j1kcdhpychv3kfozc4an5 2816011 2816010 2026-06-16T19:22:25Z Alandmanson 1669821 /* Apoidea */ 2816011 wikitext text/x-wiki =Apoidea= The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} k8la7mn3qggjrohrokv3ykj8zeiftbg 2816087 2816011 2026-06-17T07:30:12Z Alandmanson 1669821 /* Apoidea */ 2816087 wikitext text/x-wiki =Apoidea= This superfamily includes the bees and about 13 families of wasps. :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} 5h3r70lpl7n4wvovg0261sml00fwy0y 2816088 2816087 2026-06-17T07:32:43Z Alandmanson 1669821 2816088 wikitext text/x-wiki =Apoidea= This superfamily includes the bees and about 13 families of wasps. The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} l8embfdo5qjdhs2syobrexd55pta8g2 2816089 2816088 2026-06-17T07:47:19Z Alandmanson 1669821 /* Apoidea */ 2816089 wikitext text/x-wiki =Apoidea= There are many familiar species in this superfamily; it includes seven families of bees and about 13 families of wasps. <gallery mode=packed heights=200> Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} p3ishg3fv2hryjg3whikkdfz26u9ovd 2816090 2816089 2026-06-17T08:03:44Z Alandmanson 1669821 /* Apoidea */ 2816090 wikitext text/x-wiki =Apoidea= There are many familiar species in this superfamily; it includes seven families of bees and about 13 families of wasps. <gallery mode=packed heights=200> Amegilla atrocincta.jpg Xylocopa olivacea Vynbos 2.jpg Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg Ammophila ferrugineipes04.jpg Philanthus triangulum diadema 187037342.jpg </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} ojoukil018swtskjcy3x4ja3fdc45kg 2816091 2816090 2026-06-17T08:05:06Z Alandmanson 1669821 /* Apoidea */ 2816091 wikitext text/x-wiki =Apoidea= There are many familiar species in this superfamily; it includes seven families of bees and about 13 families of wasps. <gallery mode=packed heights=200> Amegilla atrocincta.jpg Xylocopa olivacea Vynbos 2.jpg Megachile maxillosa inaturalist 209496203.jpg Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg Ammophila ferrugineipes04.jpg Philanthus triangulum diadema 187037342.jpg </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} 3r4936ixuzwe8p2w60nxig7vq7uu4tr 2816092 2816091 2026-06-17T08:06:36Z Alandmanson 1669821 /* Apoidea */ 2816092 wikitext text/x-wiki =Apoidea= There are many familiar species in this superfamily; it includes seven families of bees and about 13 families of wasps. <gallery mode=packed heights=200> Amegilla atrocincta.jpg Xylocopa olivacea Vynbos 2.jpg Megachile maxillosa inaturalist 209496203.jpg Hylaeus heraldicus inaturalist 68861048.jpg Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg Ammophila ferrugineipes04.jpg Philanthus triangulum diadema 187037342.jpg </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} l886x8wf39io8ytvfhrpfasclwwvsgq 2816094 2816092 2026-06-17T09:56:23Z Alandmanson 1669821 /* Apoidea */ 2816094 wikitext text/x-wiki =Apoidea= There are many familiar species in this superfamily; it includes seven families of bees and about 13 families of wasps. <gallery mode=packed heights=200> Amegilla atrocincta.jpg|''Amegilla atrocincta'', Apidae Xylocopa olivacea Vynbos 2.jpg|''Xylocopa olivacea'', Apidae Megachile maxillosa inaturalist 209496203.jpg|''Megachile maxillosa'', Megachilidae Hylaeus heraldicus inaturalist 68861048.jpg|''Hylaeus heraldicus'', Colletidae Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg|''Sceliphron spirifex'', Sphecidae Ammophila ferrugineipes04.jpg|''Ammophila'' cf. ''ferrugineipes'', Philanthus triangulum diadema 187037342.jpg|''Philanthus triangulum'', </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} 5knuzz7eq0f98xhe6o86puqm77gmne3 2816095 2816094 2026-06-17T09:59:03Z Alandmanson 1669821 /* Apoidea */ 2816095 wikitext text/x-wiki =Apoidea= There are many familiar species in this superfamily; it includes seven families of bees and about 13 families of wasps. <gallery mode=packed heights=200> Amegilla atrocincta.jpg|''Amegilla atrocincta'', Apidae Xylocopa olivacea Vynbos 2.jpg|''Xylocopa olivacea'', Apidae Megachile maxillosa inaturalist 209496203.jpg|''Megachile maxillosa'', Megachilidae Hylaeus heraldicus inaturalist 68861048.jpg|''Hylaeus heraldicus'', Colletidae Black Mud-dauber Wasp (Sceliphron spirifex) on Buffalo-Thorn (Ziziphus mucronata) flowers ... (52739846889).jpg|''Sceliphron spirifex'', Sphecidae Ammophila ferrugineipes04.jpg|''Ammophila'' cf. ''ferrugineipes'', Sphecidae Philanthus triangulum diadema 187037342.jpg|''Philanthus triangulum'', Philanthidae </gallery> The cladogram below shows the probable relationships between the apoid wasp families (Sphecidae ''sensu lato'') and the bees (Anthophila).<ref name=Krichilsky2025>Krichilsky, E., Sann, M., & Ohl, M. (2025). Systematics of Sphecidae sensu lato: Past, Present, and Future—Quantifying Diversity, Taxonomy, and Phylogeny. Insect Systematics and Diversity, 9(6), ixaf037.</ref><ref name=waspweb>van Noort, S. 2026. WaspWeb: Hymenoptera of the World. https://www.waspweb.org/Apoidea/index.htm (accessed on 16 June 2026).</ref> {{clade| style=font-size:100%;line-height:100% |label1=[[Apoidea]] |1={{clade |1=[[Ampulicidae]] (Two Afrotropical genera in one subfamily) |2={{clade |1={{clade |1={{clade |1=Mellinidae (No Afrotropical genera) |2=[[Heterogynaidae]] (One Afrotropical genus) }} |2={{clade |1=[[Sphecidae]] (Nine Afrotropical genera in four subfamilies) |2=[[Crabronidae]] (46 Afrotropical genera in one subfamily) }} }} |2={{clade |1=[[Astatidae]] (Three Afrotropical genera) |2={{clade |1={{clade |1=[[Pemphredonidae]] (Seven Afrotropical genera in two subfamilies) |2={{clade |1=[[Philanthidae]] (Seven Afrotropical genera in four subfamilies) |2={{clade |1=[[Eremiaspheciidae]] (One Afrotropical species) |2=Entomosericidae (No Afrotropical genera) }} }} }} |2={{clade |1=[[Psenidae]] (Four Afrotropical genera) |2={{clade |1=[[Ammoplanidae]] (Two Afrotropical genera) |2=[[Anthophila]] (Bees - 2755 Afrotropical species in 99 genera; six families)<ref name=Eardley2010>Eardley, C., & Urban, R. (2010). Catalogue of Afrotropical bees (Hymenoptera: Apoidea: Apiformes). Zootaxa, 2455(1), 1-548.</ref> }} }} }} }} }} }} }} :'''Basal Apoidea''' <gallery mode=packed heights=200> Ampulicidae 37894270 suncana.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Ammophila (wasp)|''Ampulex'' cf. ''apicalis'']]) Dolichurus cf basuto iN 99066897 Sep 29, 2021.jpg|[[w:Ampulicidae|Ampulicidae]] - cockroach wasps ([[w:Dolichurus|''Dolichurus'' cf. ''basuto'']]) Astata iN 105162782 Nicola van Berkel.jpg|[[w:Astatidae|Astatidae]] - astatid wasps ([[w:Astata |''Astata'' sp.]]) </gallery> :'''Sphecid clade''' <gallery mode=packed heights=200> Gorytes natalensis 112517046.jpg|[[w:Bembicidae|Bembicidae]] - sand wasps ([[w:Gorytes |''Gorytes'' cf ''natalensis'']]) Tachysphex iN 250449986 2024 10 09 7305.jpg|[[African Arthropods/Crabroninae|Crabronidae]] - sand wasps ([[w:Tachysphex |''Tachysphex'' cf ''asinus'']]) Ammophila ferrugineipes Thread-waisted wasp IMG 2008s.jpg|[[w:Sphecidae|Sphecidae]] - mud daubers, digger & sand wasps (''[[w:Ammophila|Ammophila ferrugineipes]]'') </gallery> :'''Philanthid clade''' <gallery mode=packed heights=200> Polemistus braunsii iNaturalist 228280708.jpg|[[w:Pemphredonidae|Pemphredonidae]] - bee wolves and allies (''[[w:Polemistus braunsii|Polemistus braunsii]]'') Cerceris 2019 12 02 2310.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Cerceris|Cerceris]]'' sp.) Philanthus triangulum diadema 187037342.jpg|[[w:Philanthidae|Philanthidae]] - bee wolves and allies (''[[w:Philanthus triangulum|Philanthus triangulum diadema]]'') </gallery> :'''Families closely related to bees''' <gallery mode=packed heights=200> Psenini iN 1022563 i c riddell.jpg|[[w:Psenidae|Psenidae]] (Unidentified psenid wasp) Lindenius columbianus 02.jpg|[[w:Ammoplanidae|Ammoplanidae]] (''Ammoplanus salicis'', an ammoplanid wasp from New Mexico) </gallery> :'''Epifamily Anthophila (Bees)''' <gallery mode=packed heights=200> A mining bee, Genus Andrena.jpg|'''[[w:Andrenidae|Andrenidae]]''' - Mining bees (''Andrena'' sp.) Peltophorum africanum 1DS-II 6699.jpg|'''[[w:Apidae|Apidae]]''' - honey, cuckoo, digger & carpenter bees (''Xylocopa caffra'') Scrapter niger 2 flowers towards Avontuur.jpg|'''[[w:Colletidae|Colletidae]]''' - membrane, plasterer & masked bees (''Scrapter niger'') Halictid Bees (Spatunomia rubra) males roosting on a branch (16602329167).jpg|'''[[w:Halictidae|Halictidae]]''' - sweat bees, flower bees (''Spatunomia rubra'') Black bee in flower (6967270401).jpg|'''[[w:Megachilidae|Megachilidae]]''' - leaf-cutting bees, mason bees Rediviva, f, south africa, side 2014-11-04-13.11.43 ZS PMax (15794500671).jpg|'''[[w:Melittidae|Melittidae]]''' - melittid bees (''Rediviva'' sp.) </gallery> :'''Apoid family with unknown affinities''' <gallery mode=packed heights=200> Heterogyna04.jpg|[[w:Heterogynaidae|Heterogynaidae]] (''Heterogyna'' sp.) </gallery> *[[Ammoplanidae]] *[[Ampulicidae]] *[[Astatidae]] *[[Bembicinae|Bembicidae]] *[[Crabronidae]] *[[Entomosericidae]] *[[Eremiaspheciidae]] *[[Heterogynaidae]] *[[Mellinidae]] *[[Pemphredonidae]] *[[Philanthidae]] *[[Psenidae]] *[[Sphecidae]] Clade [[Anthophila (bee)|Anthophila]] *[[Andrenidae]] *[[Apidae]] *[[Colletidae]] *[[Halictidae]] *[[Megachilidae]] *[[Melittidae]] *[[Stenotritidae]] ==References== {{reflist}} {{BookCat}} da784txsbj4mkdtkyqc3q6if83f0yfw 6 330219 2815938 2026-06-16T13:42:36Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815938 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} 5z7u09vzj310ne8u2ywapm5dr3pfisc 6 330220 2815939 2026-06-16T13:43:21Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B simplified (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815939 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B simplified (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} obcfnlytitlq24eiqc4f6wl1enk8kxq 6 330221 2815943 2026-06-16T13:52:33Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815943 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} qixrjg72taeqodz1wfn0f2qb9jpinev 6 330222 2815948 2026-06-16T14:16:04Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815948 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260615 - 20260613) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} mz2jq8dy10pq3iscr92g1243byfbf2q 6 330223 2815952 2026-06-16T14:24:03Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815952 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} e9jfaxu1a3r0gbyghm0jshybaroadgr 6 330224 2815975 2026-06-16T16:47:10Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20260615 - 20260611) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815975 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20260615 - 20260611) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} 6xwbon1m93ydc0n7tmpph595gqhenq1 6 330225 2815977 2026-06-16T16:48:52Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2815977 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20260616 - 20260615) |Source={{own|Young1lim}} |Date=2026-06-16 |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}} mxr18lmy3cj46bo89it782p41yu6ou5 13 330226 2816000 2026-06-16T18:56:15Z Jorkin M. Penits 3094734 /* Update editing toolbar image */ new section 2816000 wikitext text/x-wiki == Update editing toolbar image == [[File:Modern Editing Toolbar for WikiPedia.png]]The image in this page of the tour that shows the editing toolbar is outdated. It would be beneficial to users on the tour to see an updated version of the toolbar, and a brief explanation of its functions. [[User:Jorkin M. Penits|Jorkin M. Penits]] ([[User talk:Jorkin M. Penits|discuss]] • [[Special:Contributions/Jorkin M. Penits|contribs]]) 18:56, 16 June 2026 (UTC) 0xl00yz3m1fua4smr7tztsojy5yyyua 2816053 2816000 2026-06-17T00:37:48Z Jtneill 10242 /* Update editing toolbar image */ Reply 2816053 wikitext text/x-wiki == Update editing toolbar image == [[File:Modern Editing Toolbar for WikiPedia.png]]The image in this page of the tour that shows the editing toolbar is outdated. It would be beneficial to users on the tour to see an updated version of the toolbar, and a brief explanation of its functions. [[User:Jorkin M. Penits|Jorkin M. Penits]] ([[User talk:Jorkin M. Penits|discuss]] • [[Special:Contributions/Jorkin M. Penits|contribs]]) 18:56, 16 June 2026 (UTC) : Thankyou Jorkin; I've [https://en.wikiversity.org/w/index.php?title=Wikiversity:Introduction/Part_2&curid=56709&diff=2816052&oldid=2324576 updated the image]. We could still add explanation of its functions. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:37, 17 June 2026 (UTC) s8uoumesxnbawm30z4a425ieiijkzby 14 330228 2816023 2026-06-16T21:56:58Z Jtneill 10242 Created page with "[[Motivation and emotion/Book/Psychotherapy]] [[Motivation and emotion/Book/Relationships]]" 2816023 wikitext text/x-wiki [[Motivation and emotion/Book/Psychotherapy]] [[Motivation and emotion/Book/Relationships]] 97si5ltj8jgj8v3whprzs8jtolf9ss2 2816024 2816023 2026-06-16T21:57:20Z Jtneill 10242 2816024 wikitext text/x-wiki [[Category:Motivation and emotion/Book/Psychotherapy]] [[Category:Motivation and emotion/Book/Relationships]] g6pzqlzxy7zrbk2pbboo1l066xsf0ez 14 330229 2816034 2026-06-16T23:41:29Z Atcovi 276019 Create. 2816034 wikitext text/x-wiki [[Category:Atcovi's Work]] [[Category:Health psychology]] ozrywoiag6zs14o5cd418r5b6rb19a3 3 330230 2816054 2026-06-17T00:38:12Z Jtneill 10242 Welcome 2816054 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Jorkin M. Penits!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. 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See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:38, 17 June 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} b2g6r9l0v2lhpbtdy1rrlnv9a2s8sei 0 330231 2816060 2026-06-17T05:23:32Z Jtneill 10242 Create 2816060 wikitext text/x-wiki List of academic journals about the psychology of motivation and emotion ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] ==Emotion== * ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] [[Category:Motivation and emotion]] 2m9vjlhtghwwl0v9jf9tw2gs5qt2fvn 2816061 2816060 2026-06-17T05:28:41Z Jtneill 10242 Expand with assistance of ChatGPT: https://chatgpt.com/share/6a323068-5190-83ec-a93b-0f296e541834 2816061 wikitext text/x-wiki List of academic journals about the psychology of motivation and emotion ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] ==Emotion== * [https://www.apa.org/pubs/journals/emo Emotion] * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] * [https://journals.sagepub.com/home/emr Emotion Review] * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] [[Category:Journals]] [[Category:Motivation and emotion]] ey5yach9molss3kamlrj2tr6b96rh0v 2816062 2816061 2026-06-17T05:31:58Z Jtneill 10242 Improve with assistance of Claude: https://claude.ai/share/b5db4204-a0bf-4bf5-9bf5-a9659cad1ccc 2816062 wikitext text/x-wiki List of academic journals about the psychology of motivation and emotion ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Journals]] [[Category:Motivation and emotion]] nf8l6jd4bhww23e6sbgtpke5av7vzwd 2816075 2816062 2026-06-17T05:53:14Z Jtneill 10242 2816075 wikitext text/x-wiki List of academic journals about the psychology of motivation and emotion. ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Journals]] [[Category:Motivation and emotion]] qudhtdlggtoe0jvs9746tlo53qbeg38 2816076 2816075 2026-06-17T05:53:51Z Jtneill 10242 2816076 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Journals]] [[Category:Motivation and emotion]] of8ppg08l9a4vdys2y666rxcd6op7yj 2816077 2816076 2026-06-17T05:55:11Z Jtneill 10242 2816077 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Psychology/Journals]] [[Category:Motivation and emotion]] pak4qdswltl6xxlkw36ci25gv3xmm1n 2816080 2816077 2026-06-17T05:56:55Z Jtneill 10242 2816080 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Motivation and emotion]] [[Category:Psychology/Journals]] lnxscuos42nv5gd2ya1wm2ll86x3b60 2816081 2816080 2026-06-17T05:57:32Z Jtneill 10242 2816081 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Motivation and emotion]] [[Category:Psychology/Journals]] fx7engt79sw350zm9b858b04klde3gw 2816082 2816081 2026-06-17T05:59:36Z Jtneill 10242 2816082 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) [[Category:Motivation and emotion]] [[Category:Psychology/Journals]] mhscvl5k0rr6exnn02tlk1o4iuatu2q 2816084 2816082 2026-06-17T06:03:38Z Jtneill 10242 + * [[w:List of psychology journals|List of psychology journals] (Wikipedia) 2816084 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) ==See also== * [[w:List of psychology journals|List of psychology journals] (Wikipedia) [[Category:Motivation and emotion]] [[Category:Psychology/Journals]] 4r27sg014j2k049snxbqbh8j4u3wkj7 2816085 2816084 2026-06-17T06:03:57Z Jtneill 10242 /* See also */ 2816085 wikitext text/x-wiki This is a list of the main academic journals which focus on the psychology of motivation and emotion. ==Motivation== * [https://www.sciencedirect.com/journal/learning-and-motivation Learning and Motivation] (Elsevier) * [https://www.apa.org/pubs/journals/mot Motivation Science] (American Psychological Association; official journal of the Society for the Science of Motivation) ==Emotion== * [https://link.springer.com/journal/42761 Affective Science] (Springer; official journal of the International Society for Research on Emotion) * [https://www.tandfonline.com/journals/pcem20 Cognition and Emotion] (Taylor & Francis) * [https://www.apa.org/pubs/journals/emo Emotion] (American Psychological Association) * [https://journals.sagepub.com/home/emr Emotion Review] (SAGE; International Society for Research on Emotion) * [https://academic.oup.com/scan Social Cognitive and Affective Neuroscience] (Oxford University Press) <!-- A section, not a journal: * [https://www.frontiersin.org/journals/psychology/sections/emotion-science Emotion Science] --> ==Motivation and emotion== * [https://link.springer.com/journal/11031 Motivation and Emotion] (Springer) ==See also== * [[w:List of psychology journals|List of psychology journals]] (Wikipedia) [[Category:Motivation and emotion]] [[Category:Psychology/Journals]] 7kzufs9idu8zmuk65op4qs39c98mfvv 10 330232 2816071 2026-06-17T05:45:23Z Jtneill 10242 Created page with "<noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the Motivation and..." 2816071 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2025 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic, sufficient chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* overuse of genAI—express more in your own words; watch out for [[w:AI slop|AI slop]] #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|genAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in edit summaries with links to the conversation sources; it appears that the feedback about the topic development in this respect has gone unheeded; if so, it violates academic integrity principles. #* I suspect there may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; if so, it violates academic integrity principles <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Add an engaging case study or scenario <!-- Overview – Scenario - Feature --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Include a relevant image <!-- Overview – Scenario - Feature --> # Scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one previous [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, there is insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Greater emphasis on systematic reviews and/or meta-analyses would be ideal # More detail about key studies would be ideal # Claims are well referenced # In some/many places, there is insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the edit summary. # Reminds the reader about the importance of the problem or phenomenon of interest # Remind the reader about the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## The target audience is international, not domestic. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Remove excessive use of bold font <!-- Written expression – Sentences --> ## Some/many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags) ## Some/many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]]. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, Studiosity, and/or peer feedback ## Check and make [https://www.grammarly.com/blog/comma/ correct use of commas] ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Secondary citations are overused; strive to consult primary sources ## Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ### If there are three or more authors, cite the first author followed by et al., then year. For example, either: #### in-text, Smith et al. (2020), or #### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ### Do not include author first name or initials ### Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ### List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ### A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ### Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ### Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ### Check and correct placement of full-stops ### Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ### Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources ## For citations, use APA style or wiki style, but not both <!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include hyperlinked dois (for 1-click access) ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of feature box(es) # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Also include links to related book chapters ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # No logged contributions }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic, sufficient chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* overuse of genAI—express more in your own words; watch out for [[w:AI slop|AI slop]] #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|genAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in edit summaries with links to the conversation sources; it appears that the feedback about the topic development in this respect has gone unheeded; if so, it violates academic integrity principles. #* I suspect there may be unacknowledged use of [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; if so, it violates academic integrity principles <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Add an engaging case study or scenario <!-- Overview – Scenario - Feature --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Include a relevant image <!-- Overview – Scenario - Feature --> # Scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one previous [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, there is insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Greater emphasis on systematic reviews and/or meta-analyses would be ideal # More detail about key studies would be ideal # Claims are well referenced # In some/many places, there is insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the edit summary. # Reminds the reader about the importance of the problem or phenomenon of interest # Remind the reader about the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## The target audience is international, not domestic. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Remove excessive use of bold font <!-- Written expression – Sentences --> ## Some/many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags) ## Some/many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]].## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, Studiosity, and/or peer feedback ## Check and make [https://www.grammarly.com/blog/comma/ correct use of commas] ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Secondary citations are overused; strive to consult primary sources ## Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ### If there are three or more authors, cite the first author followed by et al., then year. For example, either: #### in-text, Smith et al. (2020), or #### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ### Do not include author first name or initials ### Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ### List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ### A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ### Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ### Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ### Check and correct placement of full-stops ### Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ### Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources ## For citations, use APA style or wiki style, but not both<!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include hyperlinked dois (for 1-click access) ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of feature box(es) # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Also include links to related book chapters ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2025 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> gd16cp4ldaxk59tix1k06b1eva0r55v 10 330233 2816073 2026-06-17T05:47:02Z Jtneill 10242 Created page with "<noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development exercise]] for [[motivation and emotion]]. [[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==Topic development feedback== {{RoundBoxTop|theme=8}} The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the Motivation and emotion/Assessment/Top..." 2816073 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Topic|topic development exercise]] for [[motivation and emotion]]. [[Help:Transclusion|Transclude]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==Topic development feedback== {{RoundBoxTop|theme=8}} The [[Motivation and emotion/Assessment/Topic|topic development]] has been reviewed according to the [[Motivation and emotion/Assessment/Topic#Marking criteria|marking criteria]]. Written feedback is provided below, plus see the [[Motivation and emotion/Assessment/Topic/Feedback|general feedback]] page. Also check the [[Special:History/{{PAGENAME}}|page history]] for changes made whilst reviewing the plan. If you don't understand the feedback or would like further information, [[Motivation and emotion/Staff|get in touch]] to discuss. Marks are available via {{Motivation and emotion/Canvas}}. Marks are based on the latest version before the due date. {{RoundBoxBottom}} {{RoundBoxTop|theme=9}} [[File:Autoroute icone.svg|right|85px]] ===1. [[Motivation and emotion/Assessment/Topic#Title|Title]]=== {{{1|No comment}}} ===2. [[Motivation and emotion/Assessment/Topic#Headings|Headings]]=== {{{2|No comment}}} ===3. [[Motivation and emotion/Assessment/Topic#Headings|Overview]]=== {{{3|No comment}}} ===4. [[Motivation and emotion/Assessment/Topic#Key points|Key points]]=== {{{4|No comment}}} ===5. [[Motivation and emotion/Assessment/Topic#Figure|Figure]]=== {{{5|No comment}}} ===6. [[Motivation and emotion/Assessment/Topic#Learning feature|Learning feature]]=== {{{6|No comment}}} ===7. [[Motivation and emotion/Assessment/Topic#References|References]]=== {{{7|No comment}}} ===8. [[Motivation and emotion/Assessment/Topic#Resources|Resources]]=== {{{8|No comment}}} ===9. [[Motivation and emotion/Assessment/Topic#User page|User page]]=== {{{9|No comment}}} ===10. [[Motivation and emotion/Assessment/Topic#Social contribution|Social contribution]]=== {{{10|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> # |3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} ~~~~ </pre> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # |2= <!-- Headings --> #|3= <!-- Overview --> # |4= <!-- Key points--> # |5= <!-- Figure --> # |6= <!-- Learning feature --> # |7= <!-- References --> # |8= <!-- Resources --> # |9= <!-- User page --> # |10= <!-- Social contribution --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:42, 17 August 2025 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template which includes commonly used feedback comments: <pre> <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and sub-title correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or sub-title not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> # Excellent – Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Clear 2-level heading structure # Promising 2-level heading structure – could benefit from further development and/or refinement # Basic 2-level heading structure – could benefit from further development (expand) # Promising 1-level heading structure – could benefit from further development (e.g., consider using subheadings) # Basic, 1-level heading structure – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) # Under-developed, 1-level heading structure – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) # The headings lack sufficient incision into, and exposition of, the topic # Overly complicated 3-level structure – consider simplifying # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # The Overview and Conclusion should not use sub-headings # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # "Introduction" heading isn't necessary – provide this information in Overview or move into subsequent sections # Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Case study doesn't need a separate heading; instead embed case study within relevant sections # Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections # Check grammar (e.g,. missing question mark) # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest # A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest # Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest # A scenario or case study is planned # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person perspective (except 1st/2nd person can work for feature boxes/scenarios) <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Reasonably good alignment between focus questions and heading structure, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. ... <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}} – this can also be listed as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Placeholder use of quiz question(s) # Place quiz each question in the most relevant section # Focus the quiz question(s) on the take-home messages # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of table(s) # Promising use of table(s) # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Include citations for sources of information presented in the table # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Include source in brackets after link ## Use alphabetical order ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 02]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 02]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages }} ~~~~ </pre> gives <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and sub-title correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] # Title and/or sub-title not correctly worded and/or didn't use [[w:Letter case#Sentence casing|sentence casing]] (fixed) # User name removed from the page; for authorship see [[Special:History/{{PAGENAME}}|the page's edit history]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> # Excellent – Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Clear 2-level heading structure # Promising 2-level heading structure – could benefit from further development and/or refinement # Basic 2-level heading structure – could benefit from further development (expand) # Promising 1-level heading structure – could benefit from further development (e.g., consider using subheadings) # Under-developed, 1-level heading structure – develop further, perhaps using a 2-level structure for larger section(s) (i.e., including subheadings) # The headings lack sufficient incision into, and exposition of, the topic # Overly complicated 3-level structure – consider simplifying # Revise heading structure to place less emphasis on background concepts and more emphasis on the target topic (i.e., address the sub-title). The draft headings place too much emphasis on background concepts and too little on the relationship between the concepts. # Messy heading structure – needs work (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Alignment with focus questions --> # Excellent alignment between sub-title, focus questions, and heading structure # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment # Basic alignment between between sub-title, focus questions, and top-level headings. Aim to improve. # Develop closer alignment between sub-title, focus questions, and top-level headings # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with 3 to 5 sub-headings for large sections # The Overview and Conclusion should not use sub-headings # Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings # "Introduction" heading isn't necessary – provide this information in Overview or move into subsequent sections # Cover definition(s) in the Overview and/or subsequent sections with embedded inter-wiki link(s) to further information # Case study doesn't need a separate heading; instead embed case study within relevant sections # Quiz doesn't need a separate heading; instead embed quiz questions within relevant sections # Check grammar (e.g,. missing question mark) # Remove [[wikt:acronym#Noun|acronym]]s from headings # Remove citations from headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section # A scenario or case study is presented in a feature box at the start of this section; I moved an image into the feature box to help attract reader interest # A scenario or case study is presented in a feature box at the start of this section; add an image to the scenario to help attract reader interest # Move the scenario or case study into a feature box (with an image) to the start of this section to help engage reader interest # A scenario or case study is planned # Add a scenario or case study in a feature box (with an image) at the start of this section to help engage reader interest <!-- Description --> # A clear description of the problem/topic is planned or presented # A promising description of the problem/topic is planned or presented # A basic description of the problem/topic is planned or presented # Simplify/abbreviate the description of the problem/topic. Move detail into subsequent sections. # Add a brief, evocative description of the problem/topic <!-- Style --> # Use present, rather than future, tense # Use 3rd person perspective (except 1st/2nd person can work for feature boxes/scenarios) <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Reasonably good alignment between focus questions and heading structure, but consider closer alignment # Develop closer alignment between the sub-title, focus questions, and top-level headings # Use open- rather then close-ended focus questions # Use single- rather than double-barrelled focus questions # Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Solid development # Promising development # Overly broad/comprehensive; not sufficiently focused/targetted on the topic; this often occurs when genAI content is used by a prompter with insufficient background reading and understanding of the topic and/or insufficient revising and rewriting of genAI content # Basic development # Partial development # Insufficient development # No development <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # It may be that all planned aspects cannot be reasonably covered within the book chapter word count, so be selective and concentrate on key aspects that address the question in the sub-title # It is unlikely that all planned aspects can be reasonably covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title # All planned aspects cannot reasonably be covered within the book chapter word count, so be selective and concentrate on the most important aspects which address the question in the sub-title <!-- Writing style --> # The writing style is clear and easy to follow # The writing style is generally clear but could be simplified or made more concise # The writing style is difficult to follow (e.g., due to vagueness, complex wording, long sentences, long paragraphs, repetition, etc.) <!-- Theory and research --> # Good balance of theory and research # Promising balance of theory and research # Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research # Too much theory. Not enough research. Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. # Select the best theories about this topic # Select the best research about this topic <!-- Citations --> # Excellent use of citations # Very good use of citations # Good use of citations # Basic use of citations # Insufficient use of citations # Non-peer-reviewed sources should be moved to the "External links" section # Tip: Rather than starting with an author name and citation, start with the more interesting part (i.e., the substance) and put the citation at the end or mid-way through the sentence <!-- Citation style --> # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] (e.g., do not include author initials) # Use [https://apastyle.apa.org/style-grammar-guidelines/citations/basic-principles APA style 7th edition for citations] with three or more authors (i.e., FirstAuthor et al., year) # [https://apastyle.apa.org/style-grammar-guidelines/punctuation/serial-comma APA style uses serial commas][[w:Serial comma|1]][https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should 2][https://www.youtube.com/watch?v=gBx8ooDupXY 3] (1 min) <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title. # Direct quotes need page numbers (APA style) – even better, express the idea in your own words # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] # Use [https://www.aresearchguide.com/write-in-third-person.html 3rd person perspective], although a case study or feature box could use 1st or 2nd person perspective # Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour) # Move references into the References section. Keep citations in the main body. # Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression and to check grammatical and spelling errors <!-- GenAI ---> # Well done on acknowledging genAI use in the edit summary. Also share link(s) to the conversation, as per the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]]. # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is well developed # Conclusion is well underway # Conclusion is underway # Conclusion is underdeveloped # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Excellent - Relevant figure(s) presented, captioned, and cited # Relevant figure(s) are presented and captioned # Relevant figure(s) are presented # The relevance of the figure to the topic is unclear # A relevant figure is not presented and cited (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Figure caption(s) provide(s) a reasonably clear description that is connected with the main text # Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. ... <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view # Consider decreasing image size(s) to make them less dominant <!-- Creation --> # Well done on creating and uploading your own image! {{smile}} – this can also be listed as a social contribution |6= <!-- Learning feature --> <!-- Interwiki links ---> # Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Excellent use of scenarios/examples/case studies # Promising use of scenarios/examples/case studies # Keep scenarios brief # Basic use of scenario/example/case study # Placeholder use of scenarios/examples/case studies # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Excellent use of quiz question(s) # Promising use of quiz question(s) # Placeholder use of quiz question(s) # Place quiz each question in the most relevant section # Focus the quiz question(s) on the take-home messages # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Excellent use of table(s) # Promising use of table(s) # Use APA style for table captions # Add table caption # Cite each table at least once in the text # Include citations for sources of information presented in the table # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent # Very good # Good # Basic # Insufficient # To be developed <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses # At least one relevant systematic review and/or meta-analysis has been identified # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Move --> # Move Wikipedia links to the "See also" section # Move non-academic / non-peer reviewed sources to the "External links" section <!-- Citations --> # All references need in-text citation # All citations need to be in the References # Only include references which have been accessed and read <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## alphabetical order ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links ## include hyperlinked dois ## page numbers should be separated by an en-dash (–) rather than a hyphen (-) # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic # Use APA style or wiki referencing style, but not both (currently, a mixture of referencing styles is used # Don't cite AI-generated content because it is unreliable and not peer-reviewed. Instead, follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]] which include acknowledging and linking to genAI use in edit summaries, otherwise it is a violation of academic integrity. |8= <!-- Resources --> <!-- See also --> # See also ## Excellent ## Very good ## Good ## Basic ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ### Also link to relevant [[w:|Wikipedia]] pages ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Excellent ## Very good ## Good ## Basic ## One of two required external links provided ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Move Wikipedia link(s) to the "See also" section ## Move academic sources into the "References" sections and provide in-text citation ## Only include links directly related to the sub-title ## Target an international audience; Australians only represent 0.33% of the world population ## Good choice of links, but poorly formatted (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) ## Include source in brackets after link ## Use alphabetical order ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent # Used effectively # Very good # Good # Basic # Not created – see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]] <!-- Description about self --> # Excellent description about self provided # Description about self provided # Brief description about self – consider expanding # Very brief description about self – consider expanding # Add description about self <!-- Links to profile(s) --> # Link(s) provided to professional profile(s) # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided # Rename the link to the book chapter to make it more user-friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) # Add link to book chapter |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: # One out of three types of contributions made with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* comments on the [[Help:Talk page|talk page]]s of other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on the {{Motivation and emotion/Canvas}} discussion forum # None summarised on user page with direct link(s) to evidence (see [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 02]]). Looking ahead to the book chapter, see [[Motivation and emotion/Assessment/Chapter#Socialcontribution|social contributions]]. # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 02]]. # Are these contributions based on AI-generated content? If so, please follow the [[Motivation and emotion/Assessment/Using generative AI|using genAI guidelines]], otherwise it is a violation of academic integrity. # Well done on creating and uploading your own image! # Use a numbered list (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 02]]) # Descriptions of contributions could be more precise/accurate/detailed # Add a brief summary of each contribution # Remember to sign comments on talk pages }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:42, 17 August 2025 (UTC) ==See also== * [[Motivation and emotion/Assessment/Topic|Topic development guidelines]] * [[Template:MEBF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Topic]] </noinclude> rxhsos49kawdj58xm1zleqcb904wz5a 14 330234 2816079 2026-06-17T05:56:19Z Jtneill 10242 Created page with "[[Category:Journals]] [[Category:Psychology]]" 2816079 wikitext text/x-wiki [[Category:Journals]] [[Category:Psychology]] rqciwgl15adn7zgv9d9lb2y412bl2kp 6 330235 2816097 2026-06-17T10:59:39Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20260617 - 20260616) |Source={{own|Young1lim}} |Date=2026-06-17 |Author=Young W. 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